AN  INTRODUCTORY  TREATISE 

ON 

DYNAMICAL  ASTRONOMY 


CAMBRIDGE  UNIVERSITY   PRESS 

C.  F.  CLAY,  MANAGER 
LONDON  :  FETTER  LANE,  B.C.  4 


NEW  YORK  :  G.  P.  PUTNAM'S  SONS 
BOMBAY,  CALCUTTA,  MADRAS  :   MACMILLAN  AND  CO.,  LTD. 

TORONTO  :  J.  M.  DENT  AND  SONS,  LTD. 
TOKYO  :  THE  MARUZEN-KABUSHIKI-KAISHA 


All  rights  reserved 


AN  INTRODUCTORY  TREATISE 

ON 

DYNAMICAL  ASTRONOMY 


BY 


H.  C.  PLUMMER,  M.A. 
«\ 

ANDREWS  PROFESSOR  OF  ASTRONOMY  IN  THE  UNIVERSITY  OF  DUBLIN  AND 
ROYAL  ASTRONOMER  OF  IRELAND 


I- 


PEEFACE 

JL  HIS  book  is  intended  to  provide  an  introduction  to  those  parts  of  Astronomy 
which  require  dynamical  treatment.  To  cover  the  whole  of  this  wide  sub- 
ject, even  in  a  preliminary  way,  within  the  limits  of  a  single  volume  of 
moderate  size  would  be  manifestly  impossible.  Thus  the  treatment  of  bodies 
of  definite  shape  and  of  deformable  bodies  is  entirely  excluded,  and  hence  no 
reference  will  be  found  to  problems  of  geodesy  or  the  many  aspects  of  tidal 
theory,  Already  the  study  of  stellar  motions  is  bringing  the  methods  of 
statistical  mechanics  into  use  for  astronomical  purposes,  but  this  development 
is  both  too  recent  and  too  distinct  in  its  subject-matter  to  find  a  place  here. 

Nevertheless  the  book  covers  a  wider  range  of  subject  than  has  been 
usual  in  works  of  the  kind.  Thereby  two  advantages  may  be  gained.  For 
the  reader  is  spared  the  repetition  of  very  much  the  same  introductory  matter 
which  would  be  necessary  if  the  different  branches  of  the  subject  were  taken 
up  separately.  But  in  the  second  place,  and  this  is  more  important,  he  will 
see  these  branches  in  due  relation  to  one  another  and  will  realize  better  that 
he  is  dealing  not  with  several  distinct  problems  but  with  different  parts  of 
what  is  essentially  a  single  problem.  In  an  introductory  work  it  therefore 
seemed  desirable  to  make  the  scope  as  wide  as  was  compatible  with  a  reason- 
able unity  of  method,  the  more  so  on  account  of  the  almost  complete  absence 
of  similar  works  in  the  English  language. 

The  first  six  chapters  are  devoted  to  preliminary  matters,  chiefly  connected 
with  the  undisturbed  motion  of  two  bodies.  These  are  followed  by  five 
chapters  VII  to  XI  dealing  with  the  determination  of  orbits.  This  section  is 
intended  to  familiarize  the  reader  with  the  properties  of  undisturbed  motion 
by  explaining  in  general  terms  the  most  important  and  interesting  applica- 
tions. It  is  in  no  sense  complete  and  is  not  intended  to  replace  those  works 
which  are  entirely  devoted  to  this  subject.  Otherwise  it  would  have  been 
necessary  to  describe  in  detail  such  admirably  effective  methods  as  Professor 
Leuschner's  and  to  include  fully  worked  numerical  examples.  Here,  as  else- 
where, the  aim  has  been  to  give  such  an  account  of  principles  as  will  be 


vi  Preface 

instructive  to  the  reader  whose  studies  in  this  branch  go  no  further,  and  at 
the  same  time  one  which  will  help  the  student  to  understand  more  easily 
the  technical  details  to  be  met  with  in  more  special  treatises.  Though  the 
actual  details  of  practical  computation  are  entirely  excluded,  the  fact  that  all 
such  methods  end  in  numerical  application  has  by  no  means  been  overlooked. 
A  distinct  effort  has  been  made  to  leave  no  formulae  in  a  shape  unsuitable 
for  translation  into  numbers.  The  student  who  feels  the  need  will  have  no 
difficulty  in  finding  forms  of  computation  in  other  works.  At  the  same  time 
the  reader  who  will  take  the  trouble  to  work  out  such  forms  for  himself  will 
be  rewarded  with  a  much  truer  mastery  of  the  subject,  though  he  should  not 
disdain  what  is  to  be  learnt  from  the  tradition  of  practical  computers. 

An  outline  of  the  Planetary  Theory  is  given  in  the  seven  chapters  XII  to 
XVIII.  The  first  of  these  deals  exclusively  with  the  abstract  dynamical 
principles  which  are  subsequently  employed.  It  is  hoped  that  this  synopsis 
will  prove  useful  in  avoiding  the  necessity  for  frequent  reference  to  works  on 
theoretical  mechanics.  The  reader  to  whom  the  methods  are  unfamiliar  and 
who  wishes  to  become  more  fully  acquainted  with  them  may  be  referred  to 
Professor  Whittaker's  Analytical  Dynamics,  where  he  will  also  find  an  intro- 
duction to  those  more  purely  theoretical  aspects  of  the  Problem  of  Three 
Bodies  which  find  no  place  here.  To  those  who  are  familiar  with  these 
principles  in  their  abstract  form  only  the  concrete  applications  in  the  follow- 
ing chapters  may  prove  interesting.  A  chapter  on  special  perturbations  is 
included.  Here,  as  in  the  determination  of  orbits,  the  need  for  numerical 
examples  may  be  felt.  To  have  inserted  them  would  have  interfered  too 
much  with  the  general  plan  of  the  book,  and  they  will  be  found  in  the  more 
special  treatises.  But  it  was  felt  that  the  subject  could  not  be  omitted 
altogether,  and  a  concise  and  fairly  complete  account  of  the  theory  has  there- 
fore been  given.  It  may  seem  curious  that  with  the  development  of 
analytical  resources  the  need  for  these  mechanical  methods  becomes  greater 
rather  than  less,  but  so  it  is. 

Chapter  XIX  on  the  restricted  problem  of  three  bodies  is  intended  as  an 
introduction  to  the  Lunar  Theory  contained  in  Chapters  XX  and  XXI.  The 
division  of  these  two  chapters  is  partly  arbitrary,  for  the  sake  of  preserving  a 
fair  uniformity  of  length,  but  it  coincides  roughly  with  the  distinction 
between  Hill's  researches  and  the  subsequent  development  by  Professor 
Brown.  In  the  second  a  low  order  of  approximation  is  worked  out,  and  it  is 
hoped  that  this  will  serve  to  some  extent  the  double  purpose  of  making  the 


Preface  vii 

whole  method  clearer  and  of  pointing  out  the  nature  of  the  principal  terms, 
which  are  apt  to  be  entirely  hidden  by  the  complicated  machinery  of  the 
systematic  development. 

The  rotation  of  the  Earth  and  Moon  is  discussed  in  Chapters  XXII  and 
XXIII.  The  treatment  of  precession  and  nutation  is  meant  to  be  simple 
and  practical,  and  the  opportunity  is  taken  to  add  an  account  of  the  astro- 
nomical methods  of  reckoning  time  in  actual  use.  In  the  final  chapter  of  the 
book  the  theory  of  the  ordinary  methods  of  numerical  calculation  is  explained. 
This  is  necessary  for  the  proper  understanding  of  Chapter  XVIII,  but  it  also 
bears  on  various  points  which  occur  elsewhere.  Numerical  applications  find 
no  place  in  this  work.  But  let  the  mathematical  reader  be  under  no  mis- 
apprehension. The  ultimate  aim  of  all  theory  in  Astronomy  is  seldom 
attained  without  comparison  with  the  results  of  observation,  and  the  medium 
of  comparison  is  numerical.  Hence  few  parts  of  the  theory  can  be  regarded 
as  complete  till  they  are  reduced  to  a  numerical  form.  .This  is  a  process 
which  often  demands  immense  labour  and  in  itself  a  quite  special  kind  of 
skill.  It  is  just  as  essential  as  the  manipulation  of  analytical  forms. 

Originality  in  the  wider  sense  is  not  to  be  expected  and  indeed  would 
defeat  the  object  of  the  book,  which  aims  at  making  it  easier  for  the  student 
to  read  with  profit  the  larger  and  more  technical  treatises  and  to  proceed 
to  the  original  memoirs.  A  certain  freshness  in  the  manner  of  treatment  is 
possible  and,  it  is  hoped,  will  not  be  found  altogether  wanting.  Few  direct 
references  have  been  given  as  a  guide  to  further  reading,  and  this  may  be 
regretted.  But  the  opinion  may  be  expressed  that  for  the  reader  who  is 
qualified  to  profit  by  a  work  like  the  present,  and  who  wishes  to  go  further, 
the  time  has  come  when  he  should  acquire,  if  he  has  not  done  so  already,  the 
faculty  of  consulting  the  library  for  what  he  wants  without  an  apparatus  of 
special  directions.  Sign-posts  have  their  uses,  and  the  experienced  traveller 
is  the  last  to  despise  them,  but  they  are  not  conducive  to  a  spirit  of  original 
adventure. 

Since  the  main  object  in  view  has  been  to  cover  a  wide  extent  of  ground 
in  a  tolerably  adequate  way  rather  than  to  delay  over  critical  details,  the 
absence  of  mathematical  rigour  may  sometimes  be  noticed.  Very  little 
attention  is  given  to  such  questions  as  the  convergence  of  series.  It  is  not 
to  be  inferred  that  these  points  are  unimportant  or  that  the  modern  astronomer 
can  afford  to  disregard  them.  But  apart  from  a  few  simple  cases  where  the 


viii  Preface 

reader  will  either  be  able  to  supply  what  is  necessary  for  himself,  or  would 
not  benefit  even  if  a  critical  discussion  were  added,  such  questions  are 
extremely  difficult  and  have  not  always  found  a  solution  as  yet.  It  is  pre- 
cisely one  of  the  aims  of  this  book  to  increase  the  number  of  those  who  can 
appreciate  this  side  of  the  subject  and  will  contribute  to  its  elucidation. 

The  reader  who  wishes  to  proceed  further  in  any  parts  of  the  subject  to 
which  he  is  introduced  in  this  book  will  soon  find  that  the  number  of 
systematic  treatises  available  in  all  languages  is  by  no  means  large.  He 
must  turn  at  an  early  stage  to  the  study  of  original  memoirs.  It  is  not 
difficult  to  find  assistance  in  such  sources  as  the  articles  in  the  Encyklopoidie 
der  Mathematischen  Wissenschaften,  which  render  it  unnecessary  to  give  a 
bibliography.  The  subject  is  one  which  has  received  the  attention  of  the 
majority  of  the  greatest  mathematicians  during  the  last  two  centuries  and  in 
which  they  have  found  a  constant  source  of  inspiration.  Their  works  are 
generally  accessible  in  a  convenient  collected  form. 

For  the  benefit  of  any  student  who  wishes  to  supplement  his  reading  and 
has  no  means  of  obtaining  personal  advice,  the  following  works  may  be 
specially  mentioned : 

Determination  of  Orbits  and  Special  Perturbations. 

1.  J.  Bauschinger,  Bahnbestimmung  der  Himmelskorper. 

(A  source  of  fully  worked  numerical  applications.) 

2.  Publications  of  the  Lick  Observatory,  Vol.  VII. 

(Contains  an  exposition  of  A.  0.  Leuschner's  methods.) 

Planetary  and  Lunar  Theories. 

3.  F.  Tisserand,  TraM  de  mecanique  ce'leste. 

(The  most  complete  account  of  the  classical  theories.) 

4.  H.  Poincare,  Lemons  de  mecanique  celeste. 

5.  H.  Poincare,  Methodes  nouvelles  de  mecanique  celeste. 

6.  C.  V.  L.  Charlier,  Die  Mechanik  des  Himmels. 

7.  E.  W.  Brown,  An  introductory  treatise  on  the  lunar  theory. 

(Gives  full  references  to  all  the  earlier  work  on  the  subject.) 

The  great  examples  of  the  classical  methods  in  the  form  of  practical 
application  to  the  theories  of  the  planets  are  to  be  found  in  the  works  of 
Le  Verrier  (Annales  de  VObservatoire  de  Paris),  Newcomb  (Astronomical 


Preface  ix 

Papers  of  the  American  Ephemeris)  and  Hill  (Collected  Works).  The  most 
suggestive  developments,  apart  from  the  researches  of  Poincare,  are  contained 
in  the  work  of  H.  Gyld£n  (Traite  analytique  des  orbites  absolues  des  huit 
planetes  principales)  arid  P.  A.  Hansen.  All  these  works  will  repay  careful 
study,  but  the  suggestions  are  not  to  be  taken  in  any  restrictive  sense. 

The  author  of  the  present  book  has  the  best  of  reasons  for  acknowledging 
his  debt  to  most  of  the  writers  mentioned  above  and  to  others  who  are  not 
mentioned.  Some  of  the  proof  sheets  have  been  very  kindly  read  by  the 
Rev.  P.  J.  Kirkby,  D.Sc.,  late  fellow  of  New  College,  Oxford.  Acknowledge- 
ment is  also  due  to  the  staff  of  the  Cambridge  University  Press  for  their 
care  in  the  printing.  It  is  not  to  be  hoped,  in  spite  of  every  care,  that  no 
errors  have  escaped  detection,  and  the  author  will  be  glad  to  have  such  as 
are  found  brought  to  his  notice. 

H.  C.  PLUMMER. 

DUNSINK  OBSERVATORY,  Co.  DUBLIN, 
20  February  1918. 


CONTENTS 

CHAPTER  I 

THE  LAW  OF  GRAVITATION 

SECT.  PAGE 

1, 2     Kepler's  laws 1 

3,  4     The  field  of  force  central 2 

5  Acceleration  to  a  fixed  point  for  elliptic  motion         ....  3 

6  More  general  case       ..........  4 

7  Laws  of  attraction  for  elliptic  motion.     Bertrand's  problem     .         .  5 

8  The  apsidal  angle 6 

9  Condition  for  constant  apsidal  angle 7 

10  Bertrand's  theorem  on  closed  orbits 8 

11  Summary  of  results 8 

12  Newton's  law 9 

13  Gravity  and  the  Moon's  motion 10 

14  Dimensions  and  absolute  value  of  the  constant  of  gravitation  .         .  10 


CHAPTER  II 

INTRODUCTORY   PROPOSITIONS 

15  Forces  due  to  a  gravitational  system          .         .         .         .         .         .  11 

16  Potential  of  spherical  shell 12 

17  Attraction  of  a  sphere 12 

18  Potential  of  a  body  at  a  distant  .point 13 

19  Equations  of  motion  and  general  integrals 14 

20  The  same  referred  to  the  centre  of  mass 15 

21  A  theorem  of  Jacobi 16 

22  The  invariable  plane 16 

23  Relative  coordinates  and  the  disturbing  function       .  "  .         .  17 

24  Astronomical  units    .  19 


CHAPTER  III 

MOTION  UNDER  A  CENTRAL  ATTRACTION 

25,  26     Integration  in  polar  coordinates 21 

27  The  elliptic  anomalies        .........  23 

28  Solution  of  Kepler's  equation  (tig.  1)          .         .         .         .         .         .  24 


xii  Contents 

SECT.  PAGE 

29  Parabolic  motion 26 

30  Hyperbolic  motion     ..........  26 

31,  32  Hyperbolic  motion  (repulsive  force) 27 

33  The  hodograph  (fig.  2) 30 

34  Special  treatment  of  nearly  parabolic  motion 30 


CHAPTER  IV 

EXPANSIONS  IN  ELLIPTIC  MOTION 

35  Relations  between  the  anomalies        .......  33 

36  True  and  eccentric  anomalies     ..'......  34 

37  Bessel's  coefficients 35 

38  Recurrence  formulae  ..........  36 

39-41     Expansions  in  terms  of  mean  anomaly 37 

42  Transformation  from  expansion  in  eccentric  to  mean  anomaly          .  40 

43  Cauchy's  numbers 41 

44  An  example 43 

45  Hansen's  coefficients  ...         .  - 44 

46  Convergency  of  expansions  in  powers  of  e          .....  46 

47  Expansion  of  Bessel's  coefficients 47 

CHAPTER  V 

RELATIONS  BETWEEN  TWO  OR  MORE  POSITIONS  IN  AN  ORBIT 
AND  THE  TIME 

48  Determinateness  of  orbit,  given  mean  distance  and  two  points          .  49 

49  Lambert's  theorem 50 

50  Examination  of  the  ambiguity 51 

51  Euler's  theorem 53 

52  Encke's  transformation 53 

53,  54     Lambert's  theorem  for  hyperbolic  motion 54 

55  Ratio  of  focal  triangle  to  elliptic  sector  57 

56  Ratio  to  parabolic  sector 58 

57,58     Ratio  to  hyperbolic  sector 59 

59  A  general  theorem  in  approximate  forms  ....'..  61 

60  Two  applications.     Formulae  of  Gibbs 62 

61,62     Approximate  ratios  of  focal  triangles         .....  63 


CHAPTER  VI 

THE  ORBIT  IN  SPACE 

63, 64     Definition  of  elements (55 

65  Ecliptic  coordinates 57 

66  Equatorial  coordinates 68 

67  Change  in  the  plane  of  reference        ....  69 

68  Effect  of  precession  on  the  elements  .......  70 

69  The  lociis  fetus 71 


Contents  xiii 


CHAPTER  VII 

CONDITIONS  FOR  THE  DETERMINATION  OF  AN  ELLIPTIC  ORBIT 

SECT.  PAGK 

70  Geocentric  distance  and  its  derivatives 73 

71  Derivatives  of  direction-cosines —  -       •  74 

72  Deduction  of  heliocentric  coordinates  and  components  of  velocity        .  75 

73  The  elements  determined      .  .     ' 75 

74  The  equation  in  the  heliocentric  distance 76 

75  The  limiting  curve  (fig.  3) 77 

76  The  singular  curve 

77  The  apparent  orbit.     Theorem  of  Lambert   .         .         .         .         .         .  81 

78  Theorem  of  Klinkerfues 82 

79  The  small  circle  of  closest  contact 

80  Geometrical  interpretation  of  method  .      » 


CHAPTER  VIII 

DETERMINATION   OF   AN   ORBIT.      METHOD   OF  GAUSS 

81  Data  of  the  problem 85 

82  Condition  of  motion  in  a  plane 85 

83  The  middle  geocentric  distance 86 

84  The  fundamental  equation  of  Gauss               .    •     .      (  .  87 

85  First  and  last  geocentric  distances 89 

86  First  approximation 90 

87  Treatment  of  aberration 91 

88  True  ratios  of  sectors  and  triangles 91 

89  The  solution  completed 93 


CHAPTER  IX 


90  Data  for  a  parabolic  orbit 94 

91  Condition  of  motion  in  a  plane 94 

92  Use  of  Euler's  equation 95 

93  Deduction  of  parabolic  elements 96 

94  The  second  place  as  a  test 97 

95  Method  for  circular  orbit .  98 

96  Method  of  Gauss 100 

97  Circular  elements  derived              101 


xiv  Contents 

CHAPTER  X 


ORBITS   OF   DOUBLE    STARS 

SECT. 

PAGE 

98 

Nature  of  the  apparent  orbit         

103 

99 

Application  of  projective  geometry  (fig.  4)     . 

104 

100 

Five-point  constructions  (fig.  5)    

106 

101 

Other  graphical  methods       

107 

102 

Alternative  method       .        .                 

107 

103 

Use  of  equation  of  the  apparent  orbit  

108 

104 

Elements  depending  on  the  time  ....... 

110 

105 

Special  cases  ......... 

110 

106 

Differential  corrections  ......... 

112 

107 

113 

108 

Use  of  absolute  observations         .        

113 

CHAPTER  XI 

ORBITS  OF  SPECTROSCOPIC  BINARIES 


109 

Doppler's  principle        .        .         ...... 

115 

110 

Corrections  to  the  observations     

116 

111 

Nature  of  spectroscopic  binaries    

118 

112 

The  velocity  curve  (fig.  6,  a  and  b)        . 

118 

113 

Special  points  on  the  curve   

120 

114 

Analytical  solution  for  elements    

121 

115 

Properties  of  focal  chords      .     -  .         .        .        . 

122 

116 

Properties  of  diameters          

123 

117 

Integral  properties  of  velocity  curve      ...... 

124 

118 

Differential  properties    

125 

119 

Differential  corrections  to  elements       

126 

120 

Dimensions  and  mass  functions  of  system     ..... 

126 

121 

Application  to  visual  double  stars          

127 

CHAPTER  XII 

DYNAMICAL   PRINCIPLES 

122  Lagrange's  equations 129 

123  The  integral  of  energy .         .  130 

124  Canonical  equations *     .  131 

125  Contact  transformation 132 

126  The  Hamilton- Jacobi  equation      .         . 132 

127  Variation  of  arbitrary  constants 133 

128  Hamilton's  principle 134 

129  Principle  of  least  action .  135 

130  Lagrange's  and  Poisson's  brackets 136 

131  Conditions  satisfied  by  contact  transformation 138 

132  Infinitesimal  contact  transformation 139 

133  Disturbed  motion  related  to  an  integral        .         .         .         .         .         .  140 

134  Theorem  of  Poisson       .  140 


Contents  xv 
CHAPTER  XIII 

VARIATION  OF  ELEMENTS 

SECT.  PAGE 

135  Hamilton-Jacob!  form  of  solution  for  undisturbed  motion    .         .         .  142 

136  Interpretation  of  constants   .         ........  143 

137  Lagrange's  brackets 144 

138  Poisson's  brackets 145 

139  Equations  for  the  variations — .—-      .  146 

140  Modified  definition  of  mean  longitude 147 

141  Alternative  form  of  equations  for  the  variations 148 

142  Form  involving  tangential  system  of  components          .                  .         .  149 

143  Systems  of  canonical  variables 152 

144  Delaunay's  method  of  integration 153 

145  Subsequent  transformations 155 

146  Effect  of  the  process 157 

CHAPTER  XIV 

THE  DISTURBING  FUNCTION 

147  Laplace's  coefficients      ..........  158 

148  Formulae  of  recurrence 159 

149  Newcomb's  method  of  calculating  coefficients 160 

150  Direct  calculations  required           .         .         .         .         ..         .         .  161 

151  Continued  fraction  formula 162 

152  Jacobi's  coefficients •  .                  .         .  163 

153  Partial  differential  equation  for  coefficients  ......  164 

154  Hansen's  development 166 

155  Tisserand's  polynomials 167 

156  Determination  of  constant  factors 169 

157  Symbolic  form  of  complete  development 170 

158  Newcomb's  operators  • 172 

159  Indirect  part  of  disturbing  function      .         .        .         .         .        .         .  173 

160  Alternative  order  of  development 174 

161  Explicit  form  of  disturbing  function 175 

CHAPTER  XV 

ABSOLUTE   PERTURBATIONS 

162  Orbit  in  a  resisting  medium 177 

163  Nature  of  the  perturbations 178 

164  Perturbations  of  the  first  order      ........  179 

165  Secular  and  long  period  inequalities     .         .         .                 .        .         .  180 

166  Perturbations  of  higher  orders 181 

167  Classification  of  inequalities 182 

168  Jacobi's  coordinates        .         . 184 

169  The  areal  integrals.     Elimination  of  the  nodes 185 

170  Equations  of  motion 186 

171  Equations  for  disturbed  motion 187 

172  Poisson's  theorem 188 

173  Effect  of  cornmensurability  of  mean  motions          .         .         .         .         .  190 


xvi  Contents 


CHAPTER  XVI 

SECULAR  PERTURBATIONS 

SECT.  PAGE 

174  The  disturbing  function  modified 192 

175  Form  of  expansion 193 

176  Effect  of  symmetry 195 

177,178    Explicit  form  of  secular  terms 195 

179  Orthogonal  transformation  of  variables 199 

180  Solution  for  eccentric  variables 200 

181  Solution  for  oblique  variables 202 

182  Other  forms  of  the  integrals 203 

183  Upper  limit  to  eccentricities  and  inclinations        ....  204 


CHAPTER  XVII 

SECULAR  INEQUALITIES.      METHOD  OF  GAUSS 

184  Statement  of  the  problem 

185  Attraction  of  a  loaded  ring 

186  Geometrical  relations  between  the  orbits 

187  Equation  of  the  cone 

188  The  final  quadrature 

1^9  Introduction  of  elliptic  functions  ..... 

190  Integrals  expressed  by  hypergeometric  series 

191  The  potential  in  terms  of  invariants      .... 

1 92  Transformation  of  coordinates 


CHAPTER  XVIII 

SPECIAL   PERTURBATIONS 

193  Nature  of  special  perturbations 218 

194  The  difference  table 219 

195  Formulae  of  quadratures 220 

196  Application  to  a  differential  equation 221 

197  An  example 221 

198  Method  of  rectangular  coordinates 222 

199  Equations  of  motion  in  cylindrical  coordinates      ....  224 

200  Treatment  of  the  equations 225 

201  Perturbations  in  polar  coordinates  deduced 226 

202  Equations  for  variations  in  the  elements 227 

203  Calculation  of  disturbing  forces 228 

204  Perturbations  in  the  elements 229 

205  Case  of  parabolic  orbits 230 

206  Necessary  modification  of  coefficients 231 

207  Sphere  of  influence  of  a  planet 234 


Contents  xvii 


CHAPTER  XIX 

THE   RESTRICTED   PROBLEM   OF  THREE   BODIES 

SECT.  PAGE 

208  Jacobi's  integral 236 

209  '  Tisserand's  criterion .         .  236 

210  Curves  of  zero  velocity  (fig.  7) 237 

211  Points  of  relative  equilibrium , .  239 

212  Motion  in  the  neighbourhood 241 

213  Stability  of  the  motion 242 

214  The  varied  orbit 243 

215  Elementary  theory  of  the  differential  equation 245 

216  Variation  of  the  action 247 

217  Whittaker's  theorems 248 

218  Use  of  conjugate  functions 250 

219  Applications 252 

CHAPTER  XX 

LUNAR  THEORY  I 

220  Choice  of  method 254 

221  Motion  of  Sun  denned  .         . 254 

222  Force  function  for  the  Moon         .         .         .        .         .    »    .         .        .  256 

223  Equations  of  motion 257 

224  Hill's  transformation 258 

225  Further  transformation 259 

226  Variational  curve  defined 261 

227  Equations  for  coefficients' 262 

228  More  symmetrical  form          . 263 

229  Mode  of  solution 263 

230  Polar  coordinates  deduced 265 

231  Another  treatment  of  problem       ........  265 

232  Equation  of  varied  orbit 267 

233  Hill's  determinant 268 

234  Properties  of  roots 269 

235  Development  of  associated  determinant         ......  270 

236  Adams'  determination  of  g '     .        .  272 

CHAPTER  XXI 

LUNAR  THEORY  II 

237  Small  displacements  from  variational  curve 273 

238  Finite  displacements 274 

239  Terms  of  the  first  order .         .         .  275 

240  The  variation 276 

241  First  terms  calculated .         .  277 

242  Motion  of  the  perigee .         .         .  278 

243  Principal  elliptic  term.     The  Evection 279 

244  Terms  depending  on  solar  eccentricity  .        .        .        .         .        .        .  280 


xviii  Contents 

SECT.  PAGE 

245  The  Annual  Equation    .                  ........  281 

246  The  Parallactic  Inequality 283 

247  The  third  coordinate 284 

248  Motion  of  the  node .285 

249  Further  development .         .         .  286 

250  Mode  of  treatment 287 

251  Consistency  of  equations 287 

252  Higher  parts  of  motion  of  perigee 288 

253  Definitions  of  arbitrary  constants ........  289 

254  Remaining  factors  in  the  lunar  problem 291 


CHAPTER   XXII 

PRECESSION,  NUTATION  AND  TIME 

255  Euler's  equations 292 

256  Mutual  potential  of  two  distant  masses 293 

257  The  moments  calculated 294 

258  Steady  state  of  rotation 294 

259  Equations  of  motion  for  the  axis 295 

260  Change  of  axes  for  the  Moon 296 

261  Expansions  for  elliptic  motion  introduced     .......  298 

262  Mode  of  solution 299 

263  Luni-solar  precession      .        .        .        .    •     .        .        .        .        .         .  299 

264  General  precession  (fig.  8) 300 

265  Nutation 302 

266  Nutational  ellipse 303 

267  Numerical  values  for  precession 304 

268  Results  for  nutation.    Moon's  mass 305 

269  Annual  precessions  in  R.A.  and  declination 306 

270  Sidereal  time 307 

271  Mean  time 308 

272  Tropical  year 310 

273  General  remark                                                                                            .  310 


CHAPTER   XXIII 

LIBRATION  OF  THE  MOON 

274  Cassini's  laws >  .        .        .        .  312 

275  Optical  libration 312 

276  Equations  of  motion       ....  313 

277  First  condition  of  stability .  314 

278  Libration  in  longitude 315 

279  Equations  for  the  pole 316 

280  Second  condition  of  stability 318 

281  Third  condition  for  moments  of  inertia 319 

282  Second  order  terms 320 

283  Axis  of  rotation  321 


Contents  xix 


CHAPTER   XXIV 

FORMULAE  OF  NUMERICAL  CALCULATION 

SECT.  PAOB 

284  Representation  of  a  function 323 

285  The  operators  A,  8 •    .        .         .        .        .  324 

286  Stirling's  formula 325 

287  Formula  of  Gauss 326 

288  Bessel's  formula 327 

289  Lagrange's  formula 328 

290  Mechanical  differentiation 329 

291  Inverse  operations 330 

292  The  first  integral 332 

293  The  second  integral        ....                          .  333 

294  Properties  of  Fourier's  series 333 

295  Mode  of  solution  for  coefficients    .         . 334 

296  Fundamental  formulae  .        .         .        .        .         .                '.        .  335 

297  Simplifications .  335 

298  Special  case  (s=  12) 337 

299  Property  of  least  squares 338 

300  Periodic  function  of  two  variables 339 

INDEX    .        .        .                .        .        .        .  341 


THE    LAW    OF    GRAVITATION 

1.  The  foundations  of  dynamical  Astronomy  were  laid  by  Johann  Kepler 
at  the  beginning  of  the  seventeenth  century.  His  most  important  work, 
Astronomia  Nova  (De  Motibus  Stellae  Martis),  published  in  1609,  contains 
a  profound  discussion  of  the  motion  of  the  planet  Mars,  based  on  the  obser- 
vations of  Tycho  Brahe.  In  this  work  a  real  approximation  to  the  true 
kinematical  relations  of  the  solar  system  is  for  the  first  time  revealed. 
Kepler's  main  results  may  be  summarized  thus : 

(a)  The  heliocentric  motions  of  the  planets  (i.e.  their  motions  relative  to 
the  Sun)  take  place  in  fixed  planes  passing  through  the  actual  position  of  the 
Sun.  « 

(b)  The  area  of  the  sector  traced  by  the  radius  vector  from  the  Sun, 
between  any  two  positions  of  a  planet  in  its  orbit,  is  proportional  to  the  time 
occupied  in  passing  from  one  position  to  the  other. 

(c)  The  form  of  a  planetary  orbit  is  an  ellipse,  of  which  the  Sun  occupies 
one  focus. 

These  laws,  which  were  found  in  the  first  instance  to  hold  for  the  Earth 
and  for  Mars,  apply  to  the  individual  planets.  In  a  later  work,  Harmonices 
Mundi,  published  in  1619,  another  law  is  given  which  connects  the  motions 
of  the  different  planets  together.  This  is  : 

(d)  The  square  of  the  periodic  time  is  proportional  to  the  cube  of  the 
mean  distance  (i.e.  the  semi-axis  major). 

These  deductions  from  observation  are  given  here  in  the  order  in  which 
they  were  discovered.  The  third  (c)  is  generally  known  as  Kepler's  first  law, 
the  second  (6)  as  his  second  law,  and  the  fourth  (d)  as  his  third  law.  But  the 
first  statement  is  of  equal  importance.  In  the  Ptolemaic  system  the  "  first 
inequality  "  of  a  planet,  which  represents  its  heliocentric  motion,  was  assigned 
to  a  plane  passing  through  the  mean  position  of  the  Sun.  Even  in  the 
Copernican  system  this  "  mean  position  "  becomes  the  centre  of  the  Earth's 
orbit,  not  the  actual  eccentric  position  of  the  Sun.  In  consequence  no 
astronomer  before  Kepler  had  succeeded  in  representing  the  latitudes  of  the 
planets  with  even  tolerable  success. 

p.  D.  A.  1 


2  The  Law  of  Gravitation  [CH.  i 

2.  It  is  undeniable  that  in  making  his  discoveries  Kepler  was  aided  by 
a  certain  measure  of  good  fortune.     Thus  his  law  of  areas  was  in  reality 
founded  on  a  lucky  combination  of  errors.     In  the  first  place  it  was  based  on 
the  hypothesis  of  an  eccentric  circular  orbit  and  was  later  adopted  in  the 
elliptic  theory.     In  the  second  place  Kepler  supposed  (a)  that  the  time  in  a 
small  arc  was  proportional  to  the  radius  vector,  (b)  that  the  time  in  a  finite 
arc  was  therefore  proportional  to  the  sum  of  the  radii  vectores  to  all  the 
points  of  the  arc,  (c)  that  this  sum  is  represented  by  the  area  of  the  sector. 
Both  (a)  and  (c)  are  erroneous,  and  indeed  Kepler  was  aware  that  (c)  was 
not  strictly  accurate.    Mathematically  expressed,  the  argument  would  appear 
thus: 

hdt  =  rds,       ht  =  Irds  —  2  (area  of  sector). 

Both  the  supposed  fact  and  the  method  of  deduction  are  wrong,  yet  the 
result  is  right.  But  if  it  should  be  supposed  that  Kepler  owed  his  success 
to  good  fortune  it  must  be  remembered  that  this  fortune  was  simply  the 
reward  of  unparalleled  industry  in  exhausting  the  possibilities  of  every 
hypothesis  that  presented  itself  and  in  checking  the  value  of  any  new  principle 
by  direct  comparison  with  good  observations.  It  must  also  be  remarked  that 
Tycho  Brahe's  observations  were  of  the  proper  order  of  accuracy  for  Kepler's 
purpose,  being  sufficiently  accurate  to  discriminate  between  true  and  false 
hypotheses  and  yet  not  so  refined  as  to  involve  the  problem  in  a  maze  of 
unmanageable  detail.  Another  factor  in  Kepler's  success  was  his  knowledge 
of  the  Greek  mathematicians,  in  particular  of  the  works  of  Apollonius. 

3.  Kepler  had  no  conception  of  the  property  of  inertia  and  he  was 
therefore  unable  to  make  any  progress  towards  a  correct  dynamical  view  of 
planetary  motion.     It  is  interesting  to  analyze  his  results  and  to  see  what  is 
implied  by  each  of  the  above  statements  taken  by  itself. 

According  to  the  first  statement  the  planets  move  in  a  field  of  force  which 
is  such  that  every  trajectory  is  a  plane  curve.  If  we  suppose  that  the 
acceleration  at  each  point  is  a  function  of  the  coordinates  of  the  point,  an 
immediate  deduction  can  be  made  as  to  the  nature  of  the  field  of  force.  For 
let  A,  B  be  two  points  on  a  certain  trajectory,  and  let  P  be  a  third  point  not 
in  the  plane  of  this  curve.  Then  P  can  be  joined  to  A  and  to  B  by  plane 
trajectories.  The  planes  in  which  AB,  PA  and  PB  lie  meet  in  one  point  0 
(which  may  be  at  infinity).  The  acceleration  at  A  is  in  the  plane  OAB  and 
also  in  the  plane  OAP.  Hence  it'  is  along  AO.  Similarly  the  acceleration 
at  B  is  along  BO,  and  the  acceleration  at  P  is  along  PO.  But  the  point  0 
is  determined  by  the  two  points  A  and  B.  Therefore  the  acceleration  at 
every  point  of  the  field  is  directed  towards  the  fixed  point  0,  and  the  field  of 
force  is  central  (or  parallel).  Now  the  planes  of  the  orbits  all  pass  through 
the  Sun.  Hence  the  Sun  is  the  centre  of  the  field  of  force  in  which  the 


2-5]  The  Law  of  Gravitation  3 

planets  move.     For  an  analytical  proof  of  the  general  theorem  see  Halphen 
(Comptes  Rendus,  LXXXIV,  p.  944). 

4.  To  this  the  second  statement  adds  nothing  with  regard  to  the  nature 
of  the  forces,  and  might  indeed  have  been  deduced  from  the  first.  For  it 
tells  us  that 


f  f 

\r'ide=\(xdy-ydx)  = 

J  J        . 


the  Sun  being  the  origin  of  coordinates  and  h  being  a  constant.  By  differen- 
tiation we  have 

xy  —  yx  —  h 
or 

xy  —  yx  =  0. 

Thus  yl'x  =  yjx,  which  proves  that  the.  acceleration  is  towards  the  Sun  at 
every  point,  i.e.  the  field  of  force  is  central.  Clearly  the  argument  might  be 
reversed,  and  the  law  of  areas  deduced  from  the  fact  that  the  accelerations 
are  directed  towards  a  fixed  centre,  which  has  already  been  obtained  from  the 
first  statement.  Both  this  theorem  and  its  converse  are  given  in  Newton's 
Principia,  Book  I,  Props.  I  and  II. 

5.  We  shall  now  investigate  the  law  of  acceleration  towards  a  fixed  point 
under  which  elliptic  motion  is  possible.  In  the  first  instance  it  will  not  be* 
assumed  that  the  fixed  point  is  the  focus  of  the  ellipse.  Apart  from  the 
interest  of  the  more  general  result,  this  is  the  more  desirable  because  many 
pairs  of  stars  are  known  in  the  sky  the  components  of  which  are  observed  to 
revolve  around  one  another  in  apparent  ellipses ;  but  the  plane  of  the  motion 
being  unknown  it  is  only  a  matter  of  inference  that  either  star  is  in  the  focus 
of  the  relative  orbit  of  the  other.  For  it  is  the  projection  of  the  motion  on 
a  plane  perpendicular  to  the  line  of  sight  which  is  observed.  Let  then  the 

ellipse ' 

^2       2 

a2+P~ 

be  described  freely  under  an  acceleration  to  the  fixed  point  (/,  g).  Any  point 
on  the  ellipse  can  be  represented  by  (a  cos  E,  b  sin  E).  The  angle  E  which 
is  known  in  analytical  geometry  as  the  eccentric  angle  is  called  in  Astronomy 
the  eccentric  anomaly  of  the  point.  The  accelerations  being 

-asinE.E-acosE.E2,     6  cos  E.  E  -  b  sin  E.  E* 
along  the  two  axes,  we  have 

-  a  sin  E.  E  -  a  cos  E .  E*  _  b  cos  E .  E  -  b  sin  E .  E* 

acosE—f  bsinE-g 

whence 

E__      ag^cosJE  —  6/sin  E        p  ,_, 

E     ab  —  ag  sin  E  —  bfcosE' 

1—2 


4  The  Law  of  Gravitation  [OH.  I 

This  is  an  integrable  form,  giving  immediately 

E  =  h(ab-agsmE-bfcosE)~l  .....................  (2) 

or 

abE  +  ag  cos  E  —  bfsin  E  =  h(t  —  Q 

where  h  and  t0  are  constants  of  integration.     If  we  put  h  =  aba, 

E-£smE  +  !j-cosE=n(t-t0)  .....................  (3) 

Ct  0 

and  this  may  be  considered  a  generalized  form  of  what  is  known  as  Kepler's 
equation.  By  adding  2-Tr  to  E  it  is  evident  that  2?r/M  =  T  is  the  period  of  a 
whole  revolution.  Kepler's  form  applies  when  the  motion  is  about  a  focus  of 
the  ellipse,  and  can  be  obtained  by  putting  /=  ae,  g  =  0,  so  that 

E-esmE=n(t-Q  ...........................  (4) 

This  equation  is  of  fundamental  importance.  The  point  for  which  E  =  0  is 
the  nearest  point  on  the  orbit  to  the  attracting  focus  and  is  sometimes  called 
the  pe'ricentre.  The  corresponding  time  is  t0  and  n  is  called  the  mean 
motion. 

By  (1)  and  (2)  the  components  of  the  acceleration  become 

-  .  ab(f-acoBE)h* 

.E-acosE.E2  =  7—1  ----  ^—.  —  ^  —  o  -----  =- 
(ab  —  ag  sin  a  —  oj  cos  h)3 

ri     7    •     rr    ™  aft  (</  —  6  sin  E)h? 

2  ~  — 


..          ^  —  r 

(ab  —  ag  sin  A  -  67  cos 

so  that  the  total  acceleration  is  equal  to 


\     '.'••>       MS     *         ,  '  I   I    I       ,*  V       I  •    ••••!••••••••.    ••••••(O) 

a  b 

where  r  is  the  distance  of  the  point  on  the  orbit  from  (f,  g). 

6.  Before  examining  this  result  more  closely,  it  may  be  noticed  that  the 
method  is  quite  general  and  may  be  applied  to  any  central  orbit.  For  if  the 
coordinates  of  a  point  (x,  y}  on  the  curve  be  expressed  in  terms  of  a  single 
parameter  a,  we  have  similarly 

x'u  +  x"a2  _  y''d  +  y"d2 
or 

«  =    ^"  (y  -  9}  -  y"  (x  -f)  & 

where  x',  y' . . .  denote  derivatives  with  respect  to  a,  and  a,  a  derivatives  with 
respect  to  the  time.     Hence  on  integration, 

a.  =  -  h  {x'  (y-g)-  y'  (x  -f)}~1 
J  (xdy  -  y  doc)  -fy  +  gx  =  h(t- 10). 


5-7]  The  Law  of  Gravitation  5 

By  taking  the  last  integration  over  one  revolution  in  a  closed  orbit  it  is 
seen  that  h  represents  twice  the  area  divided  by  the  periodic  time.  The 
components  of  the  acceleration  become 

/) 


and  the  total  acceleration  is  therefore 

R  =  h*r  (x'y"  -  x"y')  [x  (y  -  g)  -  y'  (co  -/))- 
.    =  h?r/p3p 

where  p  is  the  radius  of  curvature  at  the  point  and  p  is  the  perpendicular 
from  (f,  g)  to  the  tangent  at  the  point.  This  of  course  is  the  well-known 
expression  for  the  acceleration  towards  the  centre  of  attraction. 

The  same  orbit  will  be  described  in  the  same  periodic  time  under  the 
central  attraction  R'  to  another  point  (/',  g')  if 

R'  =  Kr'lp'3p 
that  is,  if 

R'IR=p3r'/p'3r. 

This  result  is  equivalent  to  Principia,  Book  I,  Prop,  vii,  Cor.  3. 
7.     We  now  return  to  equation  (5)  which  may  be  written 

..................  (6) 


where  q  and  qn  are  the  perpendiculars  on  the  polar  of  (f,  g)  from  the  point 
(x,  y}  on  the  orbit  and  the  centre  of  the  ellipse  respectively.  Hence  the 
ellipse  represented  by  the  general  equation 

ax2  +  2hxy  +  bf  +  2gx  +  2fy  +  1  =  0  ..................  (7) 

can  be  described  under  an  acceleration  directed  towards  the  origin  if  the 
acceleration  follows  the  law 

R  =  m2r(l  +  gx+fy)-3,     m*  =  n'&*/C3  ..................  (8) 

where  A  and  C  have  their  usual  meaning  for  the  conic  (7).  Conversely,  if  the 
law  (8)  is  given,  the  trajectory  is  always  a  conic  whatever  the  initial  conditions 
may  be.  For  (7)  is  a  possible  orbit,  and  /  and  g  are  determined  by  the  law, 
while  a,  b  and  h  are  three  arbitrary  constants  which  can  be  chosen  so  as  to 
satisfy  any  given  conditions,  such  as  the  initial  velocity  given  in  magnitude 
and  direction  at  a  particular  point. 

There  now.  arises  the  interesting  question  whether  any  other  form  of  law 
besides  (8)  exists,  for  which  the  trajectories  are  always  conies  (Bertrand's 
problem).  Let 

R  =  ro"r//(<c,  y) 


6  The  Law  of  Gravitation  [CH.  I 

be  such  a  law.     Then  if  (7)  is  to  be  an  orbit, 

must  be  satisfied  by.  the  coordinates  of  every  point  on  (7),  i.e.  this  equation 
must  be  equivalent  to  (7).     But  (7)  can  be  written  in  either  of  the  forms 

l+gx+fy  =  );(I-ax*-2hxy-by*) 


and  clearly  in  no  other  way  which  does  not  introduce  a  greater  number  of 
independent  constants  on  the  right-hand  side.  The  first  of  these  forms  gives 
an  expression  forf(x,  y)  which  is  (like  an  infinite  number  of  others)  compatible 
with  (7),  but  only  under  restricted  conditions.  For  it  fixes  the  constants  a,  b 
and  h  and  leaves  only  /  and  g  arbitrary  ;  and  these  are  not  in  general  sufficient 
in  number  to  satisfy  the  initial  conditions.  On  the  other  hand,  the  second 
form  gives  an  expression  for  the  acceleration  which  may  be  written 

R  =  mzr  (ax*  +  2(3xy  +  yy2)~*     .....................  (9) 


This  only  requires  the  constants  in  (7)  to  satisfy  the  two  relations 


and  thus  three  other  relations  can  be  satisfied  which  are  required  by  the 
initial  conditions.  Hence  motion  under  a  central  acceleration  given  by  (9) 
is  always  in  a  conic  which  by  the  two  relations  found  touches  the  lines  (real 
or  imaginary) 


The  laws  (8)  and  (9)  are  the  only  ones  under  which  a  conic  is  always 
described  in  a  given  plane  whatever  the  initial  conditions  may  be.  Their 
character  was  first  established  by  Darboux  and  by  Halphen  (Comptes  Rendus, 
LXXXIV,  pp.  760,  936  and  939). 

8.     A  point  on  a  central  orbit  at  which  the  motion  is  at  right  angles  to 

fjrV* 

the  radius  vector  is  called  an  apse.     At  such  a  point  -^  =  0  and  the  radius 

vector  is  in  general  either  a  maximum  or  a  minimum.  Since  the  motion  is 
reversible  the  radius  vector  to  an  apse  is  an  axis  of  symmetry  in  the  orbit 
and  the  next  apsidal  distances  on  either  side  are  equal.  There  can  be  there- 
fore only  two  distinct  apsidal  distances  recurring  alternately  and  the  angle 
between  any  two  consecutive  apses  is  constant  and  is  called  the  apsidal 
angle. 

The  differential  equation  of  a  central  orbit  is  known  to  be 

d?u  P 


7-9]  The  Law  of  (Gravitation  7 

where  u  =  l/r  and  P  is  the  force  to  the  centre.  If  we  write  P  =  v?U  the 
radius  of  a  circular  orbit  is  given  by  u  =  U/h2.  Let  the  circular  orbit  be 
slightly  disturbed,  so  that  we  may  write  u  +  x  instead  of  u,  where  u  is  con- 
stant and  x  is  so  small  that  only  the  first  power  of  x  need  be  retained.  Then 


_U^_     _uU'        T7,_dU 
"h^a    ''    U  X'         ~  du' 


If  we  put 
the  equation  becomes 


,72™ 

S+-I.-0 


and  the  solution  is 

x  =  a  cos  m  (#  —  #„). 

The  apsidal  angle  is  therefore 

K=7r/m=7r(l-uU'/U)^    .....................  (10) 

For  example,  if  P  =  /j,rp,   U  =  ^u~p~z  and 


This  result  is  given  in  the  Prindpia,  Book  I,  Prop.  XLV,  Ex.  2. 

9.  Let  us  push  the  approximation  further  in  order  to  see,  if  possible, 
under  what  conditions  the  apsidal  angle  remains  unchanged  by  a  higher 
order  of  the  increment  x.  The  equation  of  the  disturbed  circular  orbit 
becomes 


+  i^"V)     ..................  (11) 

and  we  assume  a  solution 

x  —  a0  4-  «j  cos  md  +  a2  cos  2m#  +  a3  cos  3m0. 

If  ctj  is  of  the  first  order,  a0  and  az  must  be  of  the  second  order  at  least, 
and  it  will  become  clear  that  a3  is  of  the  third  order.  Hence 

x1  —  ictj2  +  (2a0a!  +  ttjtta)  cos  mO  +  |  -a*  cos  2md  +  a^  cos  3mO 
ar3  =  f  aj3  cos  mO  +  \a^  cos  3  mO. 

All  terms  of  order  higher  than  the  third  have  been  omitted  and  products 
of  the  cosines  have  been  changed  into  simple  cosines  of  the  multiple  angles. 
We  now  substitute  in  (11)  and  equate  coefficients.  Thus 


0  — 

1 

4 

1 

UU"    a* 

u     01 
uU"                             ,  1   ttZP" 

/  »  J  n    rt        \nn\\                                   rt  • 

•a.2  = 

'(i3  = 

2 
1 

.    n   .(^a,  \  0,0*;  i  g.    ^  .0, 

Mf7"  a^ 

4 
1 

f7   'ai 

lif/"                       1      M?/"7 

/'f     fl         1                                                    /Y    3 

*2 

CT    -Clia2  '  24'    U       l' 

8  The  Law  of  Gravitation  [CH.  I 

The  last  of  these  equations  confirms  the  statement  that  a3  is  of  the  third 
order,  but  will  not  be  needed  here.  The  first  three  after  the  elimination  of 
a0  and  «2  give 


5  uU 
-- 


-12     U       8     U 
or 

5uU"*  +  3U'"(U-uU'):=0    .....................  (12) 

This  equation  expresses  a  necessary  condition  which  must  be  satisfied  if 
the  apsidal  angle  is  to  remain  constant  when  the  displacement  from  a  circular 
orbit  is  considered  finite. 

10.     Let  us  consider  any  closed  orbit  to  be  determined  by  a  central 
acceleration  under  a  finite  range  of  initial  velocities.     The  number  of  apses 
in  a  complete  orbit  must  be  finite  and  (10)  shows  that  ra  must  be  a  com-. 
mensurable  number.     It  must  be  a  constant  therefore,  for  otherwise  it  would 
change  discontinuously  as  u  changes  continuously.     Hence 


is  an  equation  giving  the  form  of  U,  and  the  solution  is 

U  =  kul~m\ 

But  if  all  the  orbits  are  to  be  re-entrant,  so  that  K  is  constant,  the 
equation  (12)  must  also  be  satisfied.  Hence  substituting  the  form  just 
found,  we  have 

5m4  (1  -  m2)2  +  3m4  (1  -  m4)  =  0 
or 

2m4(4-ras)(l-ras)  =  0. 

Since  K  is  finite,  m  is  not  zero  and  we  have 

1  -  m2  =  0    or    1  -  w2  =  -  3 
giving 

U  =  k          or      U  =  ku~* 
and 

R  =  &/r2      or      R  =  kr. 

Thus  we  have  Bertrand's  remarkable  theorem  (Comptes  Rendus,  LXXVII, 
p.  849)  that  these  are  the  only  laws,  expressible  as  functions  of  the  distance, 
which  always  give  rise  to  closed  orbits  whatever  the  initial  circumstances 
may  be  (within  a  certain  range).  In  these  two  cases  m  =  l  or  2  and  the 
apsidal  angle  K  =  ir  or  |  TT. 

11.  The  results  obtained  can  now  be  brought  together.  According  to 
Kepler's  law  the  planetary  orbits  are  ellipses  with  the  centre  of  attraction, 
the  Sun,  situated  in  one  focus.  The  polar  of  the  focus  being  the  corresponding 
directrix,  we  have  in  (6)  q6=-aje  and  q  =  r/e,  so  that  the  acceleration  towards 
the  Sun  is 

R~n*tfji*  .................................  (13) 

When  the  centre  of  attraction  is  an  arbitrary  point  and  it  is  merely 
known  that  the  orbits  are  ellipses,  the  acceleration  towards  the  centre  must 


9-12]  The  Law  of  Gravitation  9 

follow  one  of  the  two  laws  expressed  by  (8)  and  (9).  These  are  not  in  general 
simple  functions  of  the  distance  and  it  is  only  "by  induction  that  we  should 
infer  from  the  apparent  orbits  of  double  stars  that  these  bodies  obey  the  law 
given  by  (13).  But  the  law  (8)  provides  a  simple  function  of  the  distance, 
R  =  m2r,  when  /=<7  =  0,  in  which  case  the  centres  of  all  possible  orbits  are 
at  the  origin,  i.e.  coincide  with  the  centre  of  attraction.  Similarly  the  law  (9) 
provides  a  simple  function  of  the  distance,  R  =  w*/r2,  when  a_==  7.  and  @  =  0. 
In  this  case  every  orbit  touches  the  lines  #2  +  y2  =  0,  showing  that  the  centre 
of  attraction  at  the  origin  is  the  focus  for  every  path.  These  are  the  only 
two  laws  of  central  acceleration  which  give  rise  to  elliptic  orbits  in  general 
and  can  be  expressed  in  simple  terms  of  the  distance.  But  we  have  also 
seen  that  the  same  restriction  is  imposed  when  it  is  merely  required  that  the 
paths  shall  be  plane  closed  curves  of  any  kind.  It  is  moreover  obvious  that 
the  law  of  the  direct  distance,  which  makes  the  attraction  of  a  distant  body 
more  effective  than  that  of  a  near  one,  cannot  be  the  law  of  nature.  The 
only  alternative  is  that  the  acceleration  varies  inversely  as  the  square  of  the 
distance,  and  this  law  can  therefore  be  based  upon  these  simple  suppositions  : 

(a)  the  planets  describe  closed  paths  in  planes  passing  through  the  Sun, 

(b)  the  centripetal  acceleration  towards  the  Sun,  required  by  (a),  is  a  simple 
function  of  the  distance  and  does  not  become  infinite  when  the  distance  is 
infinite. 

12.  We  have  now  to  consider  Kepler's  law  connecting  the  periodic  times 
of  the  planets  with  their  mean  distances  from  the  Sun.  This  states,  that  T2 
varies  as  a3.  But  T  =  ZTT/H,  so  that  n-a?  is  constant  for  all  the  planets.  Hence 
by  (13)  the  acceleration  of  each  planet  towards  the  Sun  is  ///r2  where  p  is 
constant.  The  force  of  attraction  acting  on  a  planet  is  therefore  m/z/r2  where 
ra  is  the  mass  of  the  planet.  And  observation  shows  that  the  same  form  ot 
law  holds  for  the  satellites  of  any  planet,  *e.g.  the  satellites  of  Jupiter.  Thus 
not  only  does  the  Sun  attract  the  planets  but  the  planets  themselves  appear 
to  attract  their  satellites  in  the  same  way.  It  is  but  natural  to  suppose  that 
the  forces  of  attraction  in  either  case  arise  from  an  inherent  property  of  matter, 
and  that  a  stress  exists  between  the  Sun  and  a  planet,  or  between  a  planet 
and  its  satellite.  Action  and  reaction  being  equal  and  opposite,  we  must 
suppose  the  force  proportional  not  only  to  the  mass  of  the  attracted  body  but 
equally  to  the  mass  of  the  attracting  body.  We  are  thus  led  to  Newton's  law 
of  gravitation  that  the  mutual  attraction  between  two  masses  m,  m'  at 
a  distance  r  apart  is  measured  by 

Gmm'/r2 

where  G  is  an  absolute  constant,  independent  of  the  masses  or  their  distance. 
It  must  be  noticed  that  the  law  has  been  arrived  at  from  the  consideration  of 
cases  in  which  the  dimensions  of  the  bodies  are  small  in  comparison  with  the 
distances  separating  them.  But  since  the  action  in  these  cases  is  proportional 


10  The  Law  of  Gravitation  [OH.  I 

to  the  total  masses,  it  is  to  be  supposed  that  it  applies  to  the  individual 
elements  of  the  matter  composing  them.  This  is  the  true  form  of  the  law  of 
universal  gravitation.  When  it  is  a  question  of  bodies  whose  dimensions  are 
not  negligible  in  relation  to  the  distances  of  surrounding  bodies,  a  modification 
of  the  simple  statement  must  be  expected.  The  examination  of  all  conse- 
quences of  the  law  of  gravitation,  including  a  comparison  with  the  results 
of  observation,  practically  constitutes  the  complete  function  of  dynamical 
Astronomy. 

13.  Since  the  Earth  possesses  only  one  satellite,  it  is  impossible  to  verify 
Kepler's  third  law  in  our  own  system.  But  it  is  of  historic  interest  to  calcu- 
late from  the  observed  motion  of  the  Moon  the  acceleration  towards  the  centre 
of  the  Earth  which  a  body  would  have  at  the  Earth's  surface.  The  Moon's 
sidereal  period  is  27d  7h  43m  118'5  or  2360591-5  sees.  Let  a  be  the  Moon's 
mean  distance  and  6  the  radius  of  the  Earth.  The  required  acceleration  is 


The  ratio  a/6  is  6O2745  and  b  may  be  .taken  to  be  6'378  x  108cm.  The 
result  of  substituting  these  numbers  is  to  give  for  the  acceleration  989  cm./sec.2 
In  point  of  fact  the  acceleration  of  a  body  at  the  Earth's  surface  is  in  the 
mean  (jr  =  981  cm./sec.2  But  the  discrepancy  is  not  surprising.  The  Moon 
describes  its  orbit  not  only  under  the  attraction  of  the  Earth  but  also  under 
the  disturbing  influence  of  the  Sun.  Moreover  g  is  a  variable  quantity  over 
the  Earth's  surface,  owing  to  the  Earth's  rotation  and  figure.  The  above 
calculation  is  altogether  too  rough  to  give  really  comparable  results.  But  it 
suffices  to  show  that  the  quantity  is  quite  of  the  same  order  as  g,  and  to  this 
extent  supports  the  identification  of  the  force  which  retains  the  Moon  in  its 
orbit  with  that  which  in  the  case  of  terrestrial  objects  is  known  as  weight. 
As  stated,  the  point  is  of  historical  interest  because  it  presented  a  difficulty 
to  Newton  who  was  long  misled  by  adopting  erroneous  numerical  data. 

14.  The  numerical  value  of  the  constant  G  depends  upon  the  units 
adopted.  Its  dimensions  are  given  by 

G.M*L-*  =  MLT~* 
or 

G  =  M-WT-*. 

In  C.G.s.  units  it  is  the  force  between  two  particles  each  of  1  gramme 
placed  1  cm.  apart.  The  first  determination  of  the  force  in  absolute  units  by 
a  laboratory  experiment  was  made  by  Cavendish.  Several  determinations 
have  since  been  made,  of  which  perhaps  the  two  best,  those  of  C.  V.  Boys  and 
K.  Braun,  agree  in  giving 

G  =  6-658  x  10~8 

corresponding  to  5'527  for  the  mean  density  of  the  Earth  and  5'985  x  ICFgr. 
for  the  total  mass  of  the  Earth. 


CHAPTER   II 


INTRODUCTORY    PROPOSITIONS 

15.  As  we  have  seen,  the  simple  facts  of  observation  lead  us  to  assume 
that  between  two  particles  of  masses  m  and  mf  situated  at  the  points 
P(x,  y,  z)  and  P'  '  (x  ',  y  ,  z'}  there  exists  a  force  Gmm'/r*,  where  r  is  the 
distance  PP'.  Now  the  direction  cosines  of  PP'  are 

x'  —  x      y  —  y      z1  —  z 
r  r  r 

and  hence  the  components  of  the  force  acting  on  the  particle  m  are 
Gmm'  —  —         Gmm'  -  —  ~  ,     Gmm'  — 

!>•'  tv*o  .  ••• 

or 

_d_u    _du    _d_u 

doc  '         dy  '         "dz 
where 

U  =  —  Gmm'  jr. 

If  m  is  attracted  not  by  a  single  particle  mf  but  by  any  number  typified 
by  m{  at  (xi}  yiy  Zi)  the  components  of  the  total  force  are  similarly 

_dU      _d_U      _dU 

doc  '         dy  '         dz 
where 

U=  — 

It  is  evident  that  U  is  the  work  which  the  system  of  attracting  particles 
will  do  if  the  particle  m  is  moved  from  its  actual  position  by  any  path  to 
some  standard  position,  except  for  a  constant  ;  it  is  the  potential  energy  of  m 
due  to  its  position  relative  to  the  attracting  system.  If  we  put 

,     U=-mV 


V  is  called  the  potential  of  the  attracting  system  at  the  point  P.  When 
the  potential  is  known  it  is  evident  that  the  components  of  the  attraction 
can  be  easily  calculated. 


12  Introductory  Propositions  [CH.  II 

16.  The  case  of  a  homogeneous  spherical  shell  is  of  elementary  im- 
portance. Let  m  be  the  mass  per  unit  area,  a  the  radius  and  r  the  distance 
of  the  point  P  from  the  centre.  If  0  is  the  centre  of  the  sphere,  two  cones 
with  semi-vertical  angles  <f>  and  </>  +  d(f>,  each  having  its  vertex  at  0  and  OP 
as  its  axis,  will  contain  between  them  an  annulus  on  the  surface  of  the 
sphere.  The  potential  of  this  annulus  at  P  is 

d  V  =  Gm  .  2-Tra  sin  <f>  .  ad<f>/p 
where 

p*  =  r2  +  a2  -  2?-a  cos  <£ 
or 

pdp  =  ra  sin  <f>  .  dcf> 
so  that 

dV=  Gm  . 
Hence 

V  = 


where  p2  and  p^  are  the  values  of  p  at  the  ends  of  the  diameter  through  P. 

These  values  are 

p2  =  r  +  a,     p1  =  \r  —  a. 

If  r  >  a,  P!  =  r  —  a  and  p2  —  p1  =  2a  ;  if  r  <  a,  pl  =  a  —  r  and  p2  —  p\  =  2r. 
Also  the  whole  mass  of  the  shell  is  M  =  4>7rmaz.  Hence  when  P  is  a  point 
outside  the  shell 

F=  GM/r 

or  the  potential  and  the  forces  derived  from  it  are  the  same  as  if  the  whole 
mass  of  the  shell  were  concentrated  at  the  centre.     On  the  other  hand,  when 

P  is  a  point  inside  the  shell, 

V=GM/a 

or  the  potential  is  constant  and  the  forces  derived  from  it  are  zero. 

17.     From  this  elementary  proposition  follow  immediately  two  corollaries  : 

(1)  A  sphere  of  uniform  density,  or  one  composed  of  concentric  strata 
of  uniform  density,  may  be  treated,  so  far  as  its  action  at  an  external  point 
is  concerned,  as  equivalent  to  a  single  particle  of  equal  mass  placed  at  its 
centre. 

(2)  For  a  point  within  such  a  sphere,  the  sphere  may  be  divided  into 
two  parts  by  the  concentric  sphere  passing  through  the  point.     The  outer 
part  is  inoperative  and  may  be  ignored,  while  the  inner  may  be  replaced  by 
a  particle  of  equal  mass  situated  at  the  centre. 

The  heavenly  bodies  are  for  the  most  part  approximately  spherical  in 
shape,  and  though  not  uniform  in  density  their  concentric  strata  are  in 
general  fairly  homogeneous.  They  may  therefore  be  treated  in  most  cases, 
as  regards  their  action  on  other  bodies,  as  simple  particles. 

The  motion  of  a  body  within  a  sphere  may  be  illustrated  by  the  motion 
of  a  meteor  within  a  spherical  swarm,  or  of  a  star  in  a  spherical  cluster.  If 


16-is]  Introductory  Propositions  13 

the  swarm  fills  a  sphere  uniformly  the  mass  operative  at  any  point  varies  as 
the  cube  of  the  distance  from  the  centre.  Hence  the  effective  force  towards 
the  centre  varies  directly  as  the  distance.  As  another  example  it  may  be 

proved  that  if  the  density  of  a  globular  cluster  varies  as  (1  +  r-)  ~  *,  r  being  the 
distance  from  the  centre,  each  star  moves  under  a  central  attraction  varying 

/I        i  O\  —  y 

as  r  (1  +  r2)    '. 

18.  An  approximate  expression  can  be  found  for  the  potential  of  a  body 
of  any  shape  at  a  distant  point.  Let  the  origin  of  coordinates,  0,  be  taken 
at  the  centre  of  gravity  of  the  body  and  the  axis  of  x  be  drawn  through  the 
point  P,  the  distance  OP  being  r.  Let  dm  be  an  element  of  mass  at  the 
point  (x,  y,  z}.  The  corresponding  element  of  the  potential  at  P  is 

Gdm  Gdm 


{(r  —  xf  +  y2  +  z*Y      (>'2  —  2ra?  + 


r  p 


r  r       \p        \r          \p 

where  P1}  P2,  ...  are  the  functions  known  as  Legendre's  polynomials. 

The  first  terms  are  easily  obtained  by  expansion  in  the  ordinary  way,  and 


ix   _ 

' 


Hence  if  the  expansion  is  not  carried  to  terms  beyond  the  second  order, 

}    r\     ^r"1"      2r2    / 

But  if  A,  B,  G  are  the  principal  moments  of  inertia  at  0,  and  /  is  the 
moment  of  inertia  about  Ox,  since  p2  has  been  written  for  a?  +  y2  +  z'2, 


2  - #2) dm 

J 

and  since  0  is  the  centre  of  gravity, 


\xdm  =  0. 

j 

Hence 


and  we  see  that  the  potential  of  the  body  at  P  differs  from  the  potential  of  a 
particle  of  equal  total  mass  placed  at  the  centre  of  gravity  by  a  quantity 
depending  only  on  l/r3.  Except  in  a  few  cases  this  quantity  is  negligible 


14  Introductory  Propositions  [CH.  n 

in  astronomical  problems  not  only  by  reason  of  the  great  distances  which 
separate  the  heavenly  bodies  in  comparison  with  their  linear  dimensions, 
but  because  they  possess  in  general  a  symmetry  of  form  which  makes 
A  +  B  +  G  —  3/  itself  a  small  quantity. 

19.  We  see  then  that  in  general  a  system  of  n  bodies  of  finite  dimen- 
sions can  be  replaced  by  a  system  of  n  small  particles  of  equal  masses 
occupying  the  positions  of  their  centres  of  gravity.  The  total  potential 

energy  of  the  system  is 

rr /-» •£          / 

where  miy  mj  are  two  of  the  ma'sses  and  r^  their  distance  apart.  For  if  we 
start  with  any  one  of  the  particles  this  sum,  which  consists  of  ^n(n—  1) 
terms,  represents  the  potential  energy  of  a  second  in  the  presence  of  the 
first,  of  a  third  in  the  presence  of  these  two,  and  so  on.  The  equations 
of  motion  are  3w  in  number  and,  according  to  §  15,  of  the  form 

317"  dU  dU 


Now 

a/7  T-  —  T- 

£  X—.  =  ^num-  — 3  =  0  (i  ±  j\ 

^  -/*  •  ..  f*Jf  V         *   «r  / 

Hence 

or 

2 Wf Xi  =  a1}  2 miifi  =  a2 ,  2  rriiZi  =  a3 
and 

where  (x,  y,  z~)  is  the  centre  of  gravity  of  the  system.  Thus  we  have  the  six 
integrals  corresponding  to  the  fact  that  the  centre  of  gravity  moves  with 
uniform  velocity  in  a  certain  direction. 

Again,  we  have 

dU        dU 

1V  .    

.    *-,  *€     ^ 


t  j      '  ij 

Hence 


or 

and  similarly 


=  c2 


i8-2o]  Introductory  Propositions  15 

These  are  called  the  three  integrals  of  area  and  express  the  fact  that  the  sum 
of  the  areas  described  by  the  radius  vector  to  each  mass,  each  multiplied  by 
that  mass  and  projected  on  any  given  plane,  is  constant.  They  also  show  that 
the  total  angular  momentum  of  the  system  about  any  fixed  axis  is  constant. 

Finally  we  have 

^  ^/  .  dU     .  dU     .  dU\ 

Znii  (xiXi  +  yiyi  +  zt*t)  =  -  2,  (xi  —  +  y{   —  +  z{  •=-  ) 


=  -dUJdt 
whence,  on  integration, 

^mi(x?  +  yl  +  z?)  =  h-U 

i 

where  h  is  constant.     This  is  the  integral  of  energy. 

There  are  then  in  all  ten  general  integrals  for  the  motion  of  a  system  of 
particles  moving  under  their  mutual  attractions  :  and  it  is  known  that  no 
others  exist  linder  certain  limitations  of  analytical  form  (Bruns  and  Poincare). 
They  are  in  fact  simply  those  which  apply  in  virtue  of  the  absence  of  external 
forces  acting  oil  the  system. 

20.  Let  the  centre  of  gravity  (x,  y,  z)  of  the  system  be  now  taken  as  the 
origin  of  coordinates.  If  (ft,  77^,  &)  are  the  new  coordinates  of  mi, 

Xi  =  x  +  &,  yi  =  y  +  rjiy  zi  =  z+£i 
and 

=  2mii)i  =  2  m^i  =  0. 


The  equations  of  motion  become 

g__atf  dU         u         dU 

aft'  mir}i~    dn'  m^-~a£ 

where  U  is  the  same  as  before,  but  r^  is  now  given  by 

rj  -  (ft  -  &  +  (it  -  vj)2  +  (6  -  £)2- 
For  the  integrals  of  area  we  have 


{(y  +  i,t)  (I  +  &)  -(z+ 
(ijiti  -  £&)  4  (yz  -  ~zy) 
(since  'S.mtrn  =  *S,mgi  —  Sm^j  =  2m<{j  =  0) 


or 

Smi  (r)i%i-  tity)  =  d  +  (a2b3  -  a^l^nii  =  c 
and  similarly 

-  ft{<)  =  c2  +  (ttsfej  -  a1bi)/'2 


16  Introductory  Propositions  [on.  n 

» 

The  integral  of  energy  becomes 

h-U=  &mt  {(x  +  &  +  (y  +  vtY  +  (*  +  £*)"} 

=  £  2  mi  &  +  W  +  fc)  +  i  Oi2  +  «22  +  o^/Swi 


or 
where 


=  h-\  (a?  +  a22  +  a32)/2  ra{. 


21.     An  interesting  equation  involving  the  mutual  distances  of  the  masses 
can  be  deduced.     We  have 


, 

=  2  Wi&2  .  2w.j  +  2wi  .  Swjf/  -  22m;  £;  .  'Zmj 

=  22mt-.2wi|r 
with  similar  equations  for  the  other  coordinates.     Hence 

2  Wfrn^ry2  =  2,mi.  2rat-  (^i2  +  T//  +  ^2). 
It  follows  that 

flimriflR  =  2       2^i     i  f  + 


-  22 


since  ?7  is  a  homogeneous  function  of  the  coordinates  of  degree  —  1.  The 
form  of  the  result  is  due  to  Jacobi.  Now  U  is  essentially  negative.  Hence 
if  K  be  positive  the  second  derivative  of  2  wt-  •//>,•  r^-2  will  be  always  positive  and 
the  first  derivative  will  increase  indefinitely  with  the  time.  Thus  the  first 
derivative,  even  if  negative  initially,  will  become  positive  after  a  certain  time 
and  therefore  2mjWjry2  will  increase  without  limit.  This  means  that  at  least 
one  of  the  distances  will  tend  to  become  infinite.  We  see  therefore  that 
a  necessary  (but  not  sufficient)  condition  for  the  stability  of  the  system  is  that 
h'  must  be  negative. 

22.  The  angular  momenta  whose  constant  values  are  cl}  c2,  cs  are  the 
projections  on  the  coordinate  planes  of  a  single  quantity.  They  are  there- 
fore the  components  of  a  vector  which  represents  the  resultant  angular 
momentum  about  the  axis 


For  this  axis,  which  is  fixed  in  space,  the  angular  momentum  is  a  maximum. 
The  plane  through  the  origin  0  which  is  perpendicular  to  this  axis  and 
therefore  fixed  is  called  the  invariable  plane  at  0.  About  any  line  through  0 
in  this  plane  the  angular  momentum  is  zero,  and  about  any  line  through  0 


20-23]  Introductory  Propositions  17 

making  an  angle  0  with  the  invariable  axis  (1)  the  angular  momentum  is 
\J(Ci  +  c.2  +  c32)  cos  0.  The  position  of  the  invariable  plane  is  dependent  on 
the  position  of  the  chosen  origin  of  reference. 

Here  we  have  considered  the  angular  momentum  as  arising  purely  from 
the  translational  motions  of  the  bodies  treated  as  particles.  In  reality  the 
total  angular  momentum  of  the  system  includes  also  that  part  which  arises 
from  the  rotations  of  the  bodies  about  their  axes.  This  part  itself  is  constant 
if  the  system  consists  of  unconnected,  rigid,  spherical  bodies  whose  concentric 
layers  are  homogeneous.  Under  these  conditions  the  invariable  plane  at  a 
point,  as  determined  by  the  translational  motions  of  the  system  alone, 
remains  permanently  fixed.  The  conditions  hold  very  approximately  in  a 
planetary  system.  But  precessional  movements  and  the  effects  of  tidal 
friction  cause  an  interchange  between  the  rotational  and  translational  parts 
of  the  angular  momentum,  without  disturbing  the  total  amount,  and  to  this 
extent  affect  the  position  of  the  astronomical  invariable  plane  as  defined 
above. 

The  centre  of  gravity  of  the  system  may  be  taken  instead  of  an  origin 
fixed  in  space.  The  invariable  plane  is  then 

c/f+c/17  +  c.^O  ..............................  (2) 

and  this  is  the  invariable  plane  of  Laplace.  Its  permanent  fixity  is  subject 
to  the  qualifications  just  mentioned. 

A  simple  proposition  applies  to  the  motion  of  two  bodies,  namely  that 
the  planes  through  a  fixed  point  0  and  containing  the  tangents  to  the  paths 
of  the  two  bodies  intersect  the  invariable  plane  at  0  in  one  line.  This  is 
easily  seen  to  be  true.  For  the  first  plane  passes  through  the  origin,  the 
position  of  the  first  body  (xly  ylt  z^)  and  the  consecutive  point  on  its  path 
(xl  +  x1dt,  y  \  +  y\dt,  zl-\-  ^dt).  Hence  its  equation  is 


x  (2/1*1  -  2/1*1)  +  y  (*i#i  -  *i#i)  +  z  (x.y,  -  x,  yO  =  0. 
Similarly  the  equation  of  the  second  plane  is 

«  (2/2*2  -  2/2*2)   +  y  (Mg  -  *2#2)  +  Z  (#2£/2  -  #22/2)  =  °" 

The  equations  of  these  planes  together  with  that  of  the  invariable  plane 
may  therefore  be  written 

«j  =  0,     a2  =  0,     mlal  +  m2«2  =  0 
and  these  evidently  meet  in  a  common  line  of  intersection. 

23.     When  we  deal  with  the  motions  in  the  solar  system  it  is  convenient 
to  refer  them  to  the  centre  of  the  Sun  as  origin.     Let  M  be  the  mass  of  the 
Sun,  m  the  mass  of  the  planet  specially  considered  and  let  there  be  n  other 
p.  D.  A.  2 


18  Introductory  Propositions  [CH.  n 

planets,  of  which  the  typical  mass  is  m{.     Then  the  total  potential  energy  of 
the  system  is 


where  pi  is  the  distance  of  rnt  from  the  Sun,  A;  the  distance  of  wf  from  m 

and  r  the  distance  of  m  from  the  Sun,  so  that 

» 

rj  =  (Xi  -  ajT  +  (yt  -  ytf  +  (*  -  z-f 

pf  =  (xi  -  X?  +  (Vi  -  F)2  +  (*  -  ^)2 
Ar  =  ^  -  xf  +  (yi  -  7/)2  +  (Zi  -  zj 
r*   =(x-X)*  +  (y-Yy  +  (z-Z)*. 

The  equations  of  motion  of  the  Sun  are 

MX-     dU      MY--dU      MZ--dU 
~dX'  W  3Z 

and  of  the  planet  considered 

dU  -dU  W 

mx  =  —  -^—  .     my  =  —  -^—  ,     mz  —  —  ^—  . 
dx  dy  oz 

If  (£,  i),  ^)  are  the  relative  coordinates  of  the  planet, 


Hence,  if  (fi,  m,  f<)  are  the  coordinates  of  w^  relative  to  the  Sun, 


fc_     i??     1 

m  dx  +  M 


j(x-  Xj)      M(x-X} 


A«  « 

Ai3  r3 

f     (m  + 
~ 


^i-i 

r  2w 


m(X-ac)\ 

—  -  -  r  Lr 

r3 


v 


If  then  we  put 

(3) 


"i         P< 

we  have  for  the  equations  of  relative  motion 


.3  +  ........................  (4) 

and  similarly 

Ji  =  -(m  +  ^)(?.5+~  ........................  (5) 

{  —  (m  +  JOe.I  +  ........................  (6) 


23,  24]  Introductory  Propositions  19 

The  function  R  is  called  the  disturbing  function.  When,  as  in  the  solar 
system,  the  masses  of  the  planets  are  small  in  comparison  with  that  of  the 
central  body,  M,  we  see  that  the  forces  derived  from  this  function  are  small  . 
in  comparison  with  the  attraction  of  M.  Indeed  a  first  approximation  to  the 
motion  of  the  planet  considered,  which  may  now  be  called  the  disturbed 
planet,  is  obtained  by  putting  R  —  0. 

24.  A  double  star,  or  system  of  two  stars  physically  connected  and  at  the 
same  time  isolated  from  external  influences,  may  be  considered  to  nresent  a 
case  of  the  problem  of  two  bodies.  In  the  solar  system  the  disturbing  effect 
of  the  other  planets  is  always  operating.  Since,  however,  this  effect  is  small 
in  comparison  with  the  attraction  of  the  Sun  it  is  useful  to  neglect  R  and  to 
consider  the  orbit  which  a  particular  planet  would  have  if  at  a  given  instant 
the  disturbing  forces  were  removed  and  the  planet  continued  to  move.  as  part 
of  the  system  formed  by  itself  and  the  Sun  alone,  its  velocity  in  direction  and 
amount  at  the  given  instant  being  that  which  it  actually  possesses.  Such  an 
orbit  is  called  the  osculating  orbit  corresponding  to  the  given  instant.  The 
actual  orbit  from  the  beginning  will  depart  more  and  more  from  the  osculating 
orbit,  but  for  a  short  interval  of  time  the  divergence  between  the  two  will  be 
so  small  that  an  accurate  ephemeris  can  be  calculated  from  the  elements  of 
the  osculating  orbit.  The  usefulness  of  the  conception  of  the  osculating  orbit 
goes  much  deeper  than  this,  as  will  appear  later. 

Now  the  equations  (4)  to  (6)  show  that  in  the  problem  of  two  bodies,  since 
R  =  0,  the  relative  motion  is  that  which  is  determined  by  an  acceleration 
(m  +  M)G/r*  towards  the  body  M  which  is  considered  fixed.  But  by  §  11 
(13)  a  law  of  this  form  leads  to  an  elliptic  orbit  with  mean  distance  a  and 
periodic  time  T,  where 

nT  =  2-7T,     n2a3  =  (m  +  M)  G. 

We  can  now  introduce  the  usual  system  of  astronomical  units.  Provision- 
ally they  are  taken  to  be  : 

Unit  of  time  :  one  mean  solar  day. 

Unit  of  length  :  the  Earth's  mean  distance  from  the  Sun. 
Unit  of  mass  :  the  Sun's  mass. 
Corresponding  to  this  system  G  is  replaced  by  the  constant  k2,  so  that 


which  differs  little  from  the  Earth's  mean  motion.  Here  T  is  the  sidereal 
year  expressed  in  mean  solar  days  and  m  is  the  mass  of  the  Earth  expressed 
as  a  fraction  of  that  of  the  Sun.  The  numerical  values  adopted  by  Gauss 
were  : 

T  =  365-  256  3835 

m  =  1/354  710 

2—2 


20  Introductory  Propositions  [CH.  n 

which  lead  to 

k  =  0-01 7  202  098  95,     log  k  =  8-235  581  4414  -  10. 
It  may  be  useful  to  add  that 

180° .  k/ir  =  3548"-18761,     log  (180° .  kjir)  =  3-550  006  5746 
which  differs  little  from  the  Earth's  daily  mean  motion  expressed  in  seconds. 

The  number  k  is  called  the  Gaussian  constant.  The  numerical  values 
of  ra  an<j  T  on  which  it  is  based  are  no  longer  considered  accurate.  Never- 
theless it  would  cause  great  practical  inconvenience  to  adjust  the  value  of  & 
to  more  modern  values  which  themselves  could  not  be  regarded  as  final. 
Hence  it  is  agreed  to  adopt  the  above  value  of  &  as  a  definite,  arbitrary 
constant  and  to  recognize  that  the  corresponding  unit  of  length  is  only  an 
approximation  to  the  Earth's  mean  distance  from  the  Sun.  According  to 
Newcomb  the  logarithm  of  this  distance  is  O'OOO  000  013. 

It  is  also  possible  to  put  the  constant  k  =  1  by  adopting  as  the  unit  of 
time  I/A;  =  58-13244087  mean  solar  days. 

For  brevity  we  may  often  put 

p.  =  k2  (1  +  m)  =  n?a3 

in  the  case  of  a  planetary  orbit,  and  for  a  double  star 

p  =  k-  (M  +  m)  =  n-a3 

where  M,  m  are  the  masses  of  the  two  components  when  the  mass  of  the 
Sun  is  taken  as  unity. 


CHAPTER  III 

MOTION    UNDER    A    CENTRAL    ATTRACTION 

25.  If  the  attraction  of  the  Sun  alone  is  considered,  the  relative  motion 
of  any  other  body  of  spherical  shape  is  conditioned  by  the  central  acceleration 
/j*r~z,  fj,  being  a  constant  the  value  of  which  has  been  explained.  The  equations 
of  motion  expressed  in  polar  coordinates  are  : 


r'0  +  2r6  =  0. 
The  latter  equation  gives  immediately 

r*e  =  h 

where  h  is  the  constant  of  areas.  Let  v  be  the  velocity  in  the  orbit,  P  the 
perpendicular  from  the  origin  on  the  tangent  and  ty  the  angle  which  the 
tangent  makes  with  the  radius  vector.  Then 

re  P 

—  =  sin  •&  =  — 
v  r 

so  that 

Pv  =  r*6  =  h 

or  the  velocity  is  inversely  proportional  to  P.  The  result  of  eliminating  & 
from  the  equations  of  motion  is 

r  =  h?/r3  —  yw,/r2 
whence 

r  2  =  2fi/r  -  A2/r2  +  c  ..............................  (1) 

and  from  these  again 

^2  (r2)  =  2  (rr  -h  r2)  =  2/A/r  +  2c. 

The  equation  of  energy  is 

t>»  =  r  -  +  r202  =  2fi/r  +  c  ...........................  (2) 

The  geometrical  meaning  of  the  constant  c  has  yet  to  be  found. 


22  Motion  under  a  Central  Attraction  [CH.  m 

26.     From  the  second  equation  of  motion 

d  -hu*~ 

dt~  u  de 

where  u  =  1/r.     Hence  the  first  equation  of  motion  becomes 

£+-$-• 

the  integral  of  which  is 

-7)}  ...........................  (3) 


where  e  and  7  are  the  two  constants  of  integration.  But  this  is  the  polar 
equation  of  a  conic  section  of  which  the  eccentricity  is  e  and  the  focus  is  at 
the  origin.  The  semi-latus  rectum  in  this  connexion  is  more  usually  called 
the  parameter  and  denoting  it  by  p  we  have 


p  =  h?lp,    or    h  = 
Also 

.          ,  du      i 


But  by  (1)  and  (3) 

r2  =  ^  {1  -  e2  cos2  (6  -  7)}  +  c. 
Hence 


or 

c  = 

Thus  if  2a  is  the  transverse  axis  of  the  orbit,  c  —  -  p/a  for  an  ellipse,  c  =  0  for 
a  parabola  and  c  =  +  /j,/a  for  an  hyperbola.  The  equation  of  energy  (2) 
becomes  therefore 


Again,  i/r  being  the  angle  which  the  direction  of  motion  at  (r,  6)  makes 
with  the  radius  vector  (drawn  towards  the  origin), 

fig 

v  cos  i/r  =  —  r  —  —  —-  sin  (6  —  7) 

ft 

v$u\"ty-  =  rQ  =  hu  =  ^  {1  +  e  cos  (0  —  7)} 

fv 

are  the  components  of  the  velocity  along  the  radius  vector  (inwards)  and 
perpendicular  to  it.  The  form  of  these  expressions  is  to  be  noted.  For  they 
evidently  represent  (a)  a  constant  velocity  V=  p/h  —  ^(p/p)  perpendicular  to 


26,  27]  Motion  under  a  Central  Attraction  23 

the  radius  vector,  and  (6)  a  constant  velocity  eV  in  a  direction  making  an 
angle  |TT  +  ^—  7  with  the  radius  vector,  that  is,  perpendicular  to  the  transverse 
axis.  Thus  at  perihelion  the  velocity  is  7(1  4-  e}  and  at  aphelion  (in  the  case 
of  elliptic  motion)  the  velocity  is  V(\  —  e). 

Since  h  =  vr  sin  -vjr,  the  preceding  equations  may  be  written 
fj,e  sin  (0  —  7)  =  —  v*r  sin  ty  cos  i/r 
fjue  cos  (#  —  7)  =  w2r  sin2  ^  —  /"- 

giving  e  and  7  when  v  and  ty  are  given  at  (r,  6).     Thus 
/u2  (e2  -  1)  =  v*r  (v2r  -  2/z)  sin2  \/r. 

27.  In  rinding  the  relations  which  subsist  between  positions  in  an  orbit 
and  the  time  it  is  necessary  to  consider  separately  the  three  kinds  of  conic 
section.  The  closed  orbit,  or  ellipse,  will  be  discussed  first. 

The  line  9  =  7  is  drawn  from  the  pole  (the  Sun)  in  the  direction  of  peri- 
helion. The  angle  6  —  7  is  measured  from  this  line  and  is  called  the  true 
anomaly.  Let  it  be  denoted  by  w.  Then,  if  t0  is  the  time  at  perihelion, 


t  -  t0  = 

y 


h3  f  dw 


1  +  ecosw)2' 

The  corresponding  result  in  terms  of  the  eccentric  anomaly  E  has  already 
been  found  (§  5).  It  will  be  convenient  to  write  down  the  relations  between 
the  radius  vector  and  the  true  and  eccentric  anomalies  in  the  forms  which  are 
most  frequently  required.  We  have 


Hence 


x  =  r  cos  w  =  a  (cos  E  —  e} 
y  =  r  sin  w  =  a  V(l  —  e2)  sin  E. 


=  a(l-ecosE)    .....................  (5) 


1  +  e  cos  w 
r  cos2  %w  =  a(l—e)  cos2  \E 
r  sin2  $w  =  a(l+  e)  sin2  \E 

i- 


(6) 


This  last  equation  may  be  regarded  as  the  standard  form  of  the  relation 
between  w  and  E.  If  we  write  e  =  sin  <f>  (0°  <  <£  <  90°),  as  is  commonly  done, 
then 

tan  \w  =  tan  (45°  +  $<j>)  tan  \E 

tan  \E=  tan  (45°  -  |c/>)  tan  \w 


24  Motion  under  a  Central  Attraction  [CH.  in 

where  ^w  and  ^E  are  always  in  the  same  quadrant.     Also 


cos  E  —  e  e  +  cos  w 

cos  w  =  —  —  p.  .  cos  L  =  '— 

1  —  e  cos  &  I  +  e  cos  w 


-  e    sn  -  e2  sn 

sin  w  =  ~—         —  =;-  ,     sin  E  = 
1  —  e  cos  A 


and  it  readily  follows  that 

1  —  e  cos  E  '  1  +  e  cos  w 

If  now  we  employ  (5)  and  (7)  we  obtain 

_h?  f  dw 

t  —  t n  —      ; 


/j?  Jo  (1  +ecosw)2 

'p3\  [       dE         1  -  ecosE 


But  /A=w2a3  where  w  is  the  mean  motion;  the  angle  n(t  —  t0)  is  called  the 
mean  anomaly  and  may  be  denoted  by  M.  We  have  therefore  once  more 
obtained  Kepler's  equation 

M  =  n(t-t0)  =  E-esmE  ..  ......................  (8) 

the  angles  M  and  E  being  expressed  in  circular  measure  ;  or  if  M  and  E  are 
expressed  in  degrees,  e  must  also  be  converted  to  the  same  form  by  the 
factor  180°/7r. 

28.  The  complete  solution  of  the  problem  of  elliptic  motion  is  contained 
in  the  equations  given  above.  No  difficulty  in  numerical  solution  arises 
except  in  the  case  of  Kepler's  equation  when  E  is  to  be  found  for  given 
values  of  e  and  M.  The  general  method  applicable  in  such  cases  may  be 
illustrated  here.  By  some  means  an  approximate  solution  E0  is  found.  Let 
E0  +  &E0  be  the  exact  solution,  and 


M0  =  E0  —  e  sin  E0. 
Then 

M  =  M0  +  (1  -  e  cos  E0)  A.E'o  +  ... 

when  E  —  e  sin  E  is  expanded  in  a  power  series  in  &E0  by  Taylor's  theorem. 
Neglecting  higher  powers  of  &E0  we  have 

=  (M  -  M0)/(l  -  e  cos  E0) 


and  hence  a  second  approximation  E1  =  E0  +  &E0.  If  this  value  is  not 
sufficiently  accurate  the  process  may  be  repeated  until  a  satisfactory  result  is 
obtained. 


27,  28] 


Motion  under  a  Central  Attraction 


25 


In  order  to  obtain  a  good  approximate  solution  at  the  outset  a  great 
variety  of  methods  have  been  devised.  These  depend  upon  (a)  the  use  of 
special  tables,  (6)  an  approximate  formula  or  a  series,  or  (c)  a  graphical 
method.  Thus  to  the  first  order  in  e, 


EQ  =  M  +  e  sin  M 


and  to  the  second  order  in  e 


where 


tan  E0  =  sec  </>  tan  2^ 

tan  ^  =  tan  (45°  +  ^</>)  tan 
the  verification  of  which  may  be  left  as  an  exercise. 


Among  graphical  methods  we  can  refer  only  to  one,  given  by  Newton 
(Prindpia,  Book  I,  Prop.  xxxi).  Consider  a  circle  of  unit  radius  and  centre  C 
rolling  on  a  straight  line  OX.  Let  E  be  the  point  of  contact  and  A  the 
point  on  the  circumference  initially  coinciding  with  0.  Let  P  be  a  point  on 
the  radius  CA  such  that  CP  =  e  and  M  and  N  the  feet  of  the  perpendiculars 
from  P  on  OX  and  CE.  Then  if  E  =  Z  A  CE  =  arc  A  E  =  OE, 


Hence  if  the  circle  is  rolled  (without  slipping)  along  OX  until  the  point 
P  is  on  the  ordinate  PM  where  OM=M,  the  point  of  contact  gives  OE=  E, 
which  can  therefore  be  read  off  when  M  is  given.  The  locus  of  P  is  evidently 
a  trochoid.  It  may  also  be  noted  that  the  ordinate 

PM=  CE-CN  =  1-ecosE 

which  is  the  corresponding  value  of  rfa  or  of  dM/dE,  and  so  gives  the  factor 
required  for  the  improvement  of  an  approximate  value  E0.  For  references 
to  practical  applications  of  the  above  principle  see  Monthly  Notices,  R.  A.  S., 
LXVII,  p.  67. 


26  Motion  under  a  Central  Attraction  [CH.  in 

29.     In  the  case  of  parabolic  motion 

dw 


=  A  /(—  )  f  H1  +  tan2  W  d  (tan 


and  therefore  a  quantity  M  may  be  denned  by  the  relation 

...............  (9) 


A  table,  known  as  Barker's  Table,  gives  M  (or  M  multiplied  by  a  certain 
numerical  factor)  with  the  argument  w.  An  inverse  table  giving  w  with  the 
argument  M  will  be  found  in  Bauschinger's  Tafeln  (No.  xv).  Or  w  may  be 
deduced  when  t  —  t0  is  given  thus.  The  equation  (9)  may  be  compared  with 
the  identity 


Hence 

tan  \w  =  X  —  - 
X 

if 


X3' 
Let 

X  =  —  tan  7,     Xa  =  —  tan 
Then 


flif O        /  I   ™1  /V f  \ r>nt  9  /3 
.iu  —  o  «  /  I     •  I  \»  —  "ot  —  mju  ^-AJ 
V  \jpv 

tan  $  =  tan3  7 
and 

tan  ^w  =  2  cot  27. 

By  these  equations  w  can  be  calculated  directly  when  t  is  given. 

30.  Hyperbolic  motion  along  the  concave  branch  of  the  curve  under 
attraction  to  the  focus  may  be  treated  in  an  analogous  way  to  elliptic  motion 
by  using  hyperbolic  functions  instead  of  circular  functions  of  the  eccentric 
anomaly.  Thus  we  have 

x  —  r  cos  w  =  a  (e  —  cosh  F) 

y  =  r  sin  w  =  a  *J(e2  —  1 )  sinh  F 
so  that 

r=   a(e^    1)  =a(ecoshF-l) (10) 

1  +  e  cos  w 


29-31  ]  Motion  under  a  Central  Attraction  27 

r  cos2  %w  =  a  (e  —  1)  cosh2  %F 
r  sin2  \w  =  a  (e  +  1)  sinh2  $F 

F    (11) 


e  —  cosh  F  ,    ,-,      e  +  cos  w 

cos  w  =  — T— ^ — =•  ,  cosh  f  =  3— - 

e  cosh  JP  —  1  1  +  e  cos  w 

sin  w  =  —  — 1.  sinh  F  =  — 

e  co.sh  ^  —  1  1  +  e  cos  - 


—  1  1  + 

By  employing  (10)  and  (12)  we  now  obtain 


^2Jo(l 

=  V  \/rJ  J0\/(e2-l)  '        e'-l 

7-,P)    (13) 


which  is  the  analogue  of  Kepler's  equation  for  this  case. 

Analogy  suggests  the  use  of  hyperbolic  functions,  but  full  and  accurate 
tables  of  these  functions  are  not  always  available.  Hence  it  is  convenient  to 
introduce  /,  the  Gudermannian  function  of  F,  where  (Log  denoting  natural- 
logarithm) 

F=  Log  tan  (45°  +  £/) 
or 

sinh  F=  tan/,     cosh  F  =  sec/,     tanh  ^  =  tan  £/! 

We  may  also  put  e  =  seci/r.     The  principal  formulae  (10),  (11)  and  (13)  then 
become 

r  =  a(esecf—  1)     (14) 

tan  £w  =  cot^i/r  tan^-/  (15) 

and 

V(/^a~3)  (t  -t0)=e  tan/-  Log  tan  (45°  +  £/) (16) 

The  last  equation  may  also  be  written 

VO^cr3)  \(t-  <o)  =  ^e  tan/-  log  tan  (45°  + 1/) 
where  log  denotes  common  logarithm  and  log\  =  9'6377843. 

Comets  moving  in  hyperbolic  orbits  are  few  in  number,  and  in  no  case 
does  the  eccentricity  greatly  exceed  unity. 

31.  There  are  certain  astronomical  problems  which  require  the  con- 
sideration of  repulsive  forces  according  to  the  law  /j,r~2  which  are  of  the 
same  form  as  gravitational  attraction  but  differ  in  sense.  The  small  particles 
which  constitute  a  comet's  tail  are  apparently  subject  to  such  forces  and 


28  Motion  under  a  Central  Attraction  [CH.  in 

finely  divided  meteoric  matter  in  the  solar  system  must  move  under  the 
pressure  due  to  the  Sun's  radiation.  Hence  we  shall  consider  the  effect  of 
replacing  +/i,  the  acceleration  at  unit  distance,  by  —  //.  The  differential 
equation  of  the  orbit  becomes 

d2u  IM  _ 

d&+    +/7  = 

the  integral  of  which  is 


=  p~*  (e  cos  w—  1  )  ..............................  (17) 

If  we  restrict  w  to  such  a  range  of  values  that  u  (or  r)  is  positive,  this 
equation  gives  only  the  branch  of  the  hyperbola  convex  to  the  centre  of 
repulsion  at  the  focus,  just  as  under  the  same  restriction  the  equation  (10) 
gives  only  the  branch  concave  to  the  centre  of  attraction.  As  compared 
with  §  26  the  signs  of  p  and  e,  as  well  as  of  p,  have  been  changed.  Hence 
the  constant  c  in  the  equation  of  energy  becomes 


so  that  the  equation  of  energy  is  now 

v2=fjf/a-2fjL'/r  ..............................  (18) 

Also,  if  T/T  is  the  angle  which  the  direction  of  motion  at  (r,  6)  makes  with  the 
radius  vector  drawn  towards  the  origin, 

,  du         u!e   .    ,n       , 
v  cos  -Jf  =  —  r    =  /z,  -^  =  —  T-  sin  (8  —  7) 
dd          h 

t 
vsmty  =     r0  =  hu     =    ^  {ecos  (6  —  7)  —  1} 

are  the  components  of  the  velocity  along  the  inward  radius  vector  and  ' 
perpendicular  to  it.  These  are  evidently  equivalent  to  (a)  a  constant 
velocity  -V'  =  —  /jffh  =  —  V(/*'/P)  perpendicular  to  the  radius  vector,  the 
negative  sign  meaning  that  V  is  drawn  in  the  sense  opposite  to  that  in 
which  the  radius  vector  is  rotating,  and  (6)  a  constant  velocity  eV  in  a 
direction  making  an  angle  |TT  -f  0  —  7  with  the  radius  vector,  that  is,  perpen- 
dicular to  the  transverse  axis.  Thus  at  perihelion  the  velocity  is  V  (e  —  1) 
as  compared  with  the  velocity  V  (e  +  1)  at  perihelion  on  the  concave  branch 
under  an  attracting  force. 

If  the  circumstances  of  projection  are  given  in  the  form  of  v  and  ty  at  the 
point  (r,  0),  we  have 

fjfp  =  h2  =  t;2r2  sin2  i/r 

fji'e  sin  (6  —  7)  =  —  vzr  sin  ^r  cos  ty 
pe  cos  (9  —  7)  =      vzr  sin2  ty  +  /*' 

which  determine  p,  e  and  7  in  terms  of  given  quantities.     In  particular 
p-  (e2  —  1  )  =  v~r  (v"r  +  2//)  sin2  -v|r. 


si,  32]  Motion  under  a  Central  Attraction  29 

32.     Expressing  the  coordinates  in  terms  of  hyperbolic  functions  we  now 
have,  since  the  centre  is  at  (ae,  0), 

x  =  r  cos  w  =  a  (e  +  cosh  F) 

y-r  sin  w  =  a  \/(e2  —  1)  sinh  F. 
Hence 

r=aL^^L  =a(ecoshJP+l)  ......  ^  ...........  (19) 

e  cos  w  —  1 

r  cos2  ^  w  =  a  (e  +  1)  cosh2  ^ 
7^  sin2  £'«;  =  a  (e  —  1)  sirih2  %F 

jUanhp7  .................................  (20) 

T  i/ 

e  +  cosh  jP  e  -  cos  w 

cos  w  —  -  cosn  JP  =  —  —  ~ 

ecosh^'+l  •  ecosw-1 


-  .   ,    F 

sm  w  =  "  -  ' 


It  then  follows  that 

fr2  h3    f  dw 

t~to=\Jide=f  J0(ecosw-l)2 

rps\   f       dF       e  cosh  F  +  1 


/ 

V 


^ 

...(22) 


which  corresponds  to  Kepler's  equation  for  this  case. 

As  in  the  case  of  an  attracting  force  we  may  now  put 

tan|/=tanh|JPT,     sec/=  cosh  F,     tan/=sinhF 

and  e  —  sec  -^.      With  these  transformations  the  principal  formulae  of  the 
solution  become 

r  =  a  (e  sec/+  1)  ..............................  (23) 

tan  \w  =  tan  J-f  tan  \f   ...........................  (24) 

V(//a-3)  (t  -Q  =  e  tan/+  Log  tan  (45°  +  1/)  .............  (25) 

or,  as  the  last  may  be  written, 

V(//ft-3)  \(t-  t0)  =  \e  tan/+  log  tan  (45° 
in  the  notation  previously  explained. 


Motion  under  a  Central  Attraction 


[CH.  Ill 


33.  The  simple  and  important  representation  of  the  velocity  in  all  cases 
as  the  resultant  of  two  vectors  both  constant  in  magnitude,  and  one  constant 
in  direction  also,  may  be  illustrated  by  considering  the  hodograph  of  the 
motion.  This  curve  is  clearly  a  circle  of  radius  V  and  centre  at  a  distance 
eV  from  the  origin.  •  The  four  figures  given  correspond  with  the  four  distinct 
types  of  motion,  (a)  elliptic,  (b)  parabolic,  (c)  hyperbolic,  under  attraction  to 
the  focus,  and  (d)  hyperbolic,  under  repulsion  from  the  focus.  In  all  cases  0 
is  the  origin,  C  the  centre,  and  OP  represents  the  velocity  at  perihelion.  If 
Q  is  any  point  on  the  hodograph,  OQ  represents  the  velocity  in  the  orbit  at 
one  extremity  of  the  focal  chord  which  is  at  right  angles  to  CQ.  The  radius 
CP  being  V,  OC  =  e  V  and  as  the  eccentricity  increases  0  moves  along  the 
radius  opposite  to  CP  from  the  position  C  for  a  circular  orbit  to  a  point  on 
the  circumference  for  a  parabolic  orbit.  As  e  increases  beyond  the  value  1 


(a)  (b)  Fig.  2. 

the  point  0  passes  outside  the  circle.  But  the  hodograph  corresponding  to 
hyperbolic  motion  is  no  longer  a  complete  circle  since  the  possible  directions 
of  motion  are  limited  by  the  asymptotes.  If  OA,  OB  are  the  tangents  from  0 
to  the  circle  the  angles  CO  A,  COB  are  each  equal  to  sin"1  e~l  and  it  is  easily 
seen  that  OA,  OB  are  parallel  to  the  asymptotes  of  the  orbit,  that  AOB  is 
equal  to  the  exterior  angle  between  the  asymptotes,  and  that  the  arc  APB 
constitutes  the  whole  hodograph.  When  the  attraction  is  changed  to  a 
repulsion  and  motion  takes  place  along  the  convex  instead  of  the  concave 
branch  of  the  hyperbola,  OP  =  V  (e—  1),  and  the  hodograph  is  confined  to 
that  arc  of  the  circle  which  is  at  all  points  convex  to  0,  whereas  in  case  (c) 
it  was  everywhere  concave  to  0. 

34.     From  the  point  of  view  of  practical  calculation  there  are  points  con- 
nected with  orbits  nearly  parabolic  in  form  which  require  special  attention. 
Kepler's  equation  for  elliptic  motion  may  be  written 
M=  E-  sin  E  +  (1  -  e)  sin  E. 

When  1  —  e  is  small  the  accurate  calculation  of  M  depends  on  that  of 
E  —  sin  E.  But  if  E  is  small  the  latter  expression  is  the  difference  of  two 
nearly  equal  quantities  and  cannot  be  calculated  directly,  unless  each  is 


3.3,  34]  Motion  under  a  Central  Attraction  31 

expressed  by  a  disproportionate  number  of  significant  figures.  Hence  the 
need  for  special  tables  (e.g.  Bauschinger's  Tafeln,  No.  XL)  or  an  approximate 
formula.  Under  the  latter  head  may  be  mentioned  the  function 


which  is  so  close  an  approximation  to  E  —  sin  E  over  the  range  of  E  from 
0°  to  70°  that  the  logarithms  of  the  two  expressions  never  differ  by  more  than 
2  in  the  seventh  place. 

It  is  evident  that  in  the  parabola  itself  E  is  evanescent  and  generally  in 
the  ellipse  of  great  eccentricity  E  is  small  at  all  points  near  the  attracting 
focus.  The  method  given  by  Gauss  in  the  Theoria  Motus  for  the  treatment 
of  Kepler's  equation  is  a  particularly  instructive  example  of  the  construction 
and  use  of  special  tables  and  as  at  the  same  time  it  brings  out  clearly  the 
relation  to  parabolic  motion  its  principle  will  be  explained  here. 

Kepler's  equation  may  be  written  in  the  form 

M  =    (l-e)(<*E  +  /3  sin  E)  +  (j3  +  ae)(E-  sin  E) 
if  «  +  j3  =  1,  or 

M=    (l-6).2A*B  +  (0  +  <K).$A*B  .....................  (26) 

if 

4=8  (E  -  sin  E)/2  (aE  +  /3  sin  E) 
and 

£2  =     (aE  +  /3  sin  E)sfQ  (E  -  sin  E) 


which  differs  from   unity  by  a  quantity  of  the  fourth  order  only  in  E  if 
/3  =  1/10,  a  =  9/10.     With  these  values  it  is  readily  found  that 


Hence  log  B  is  a  small  quantity  of  the  fourth  order  which  is  tabulated  with  A, 
itself  of  the  second  order,  as  argument. 

We  now  put,  in  view  of  (26), 

.j         //5-5e 

- 


so  that 

M  =  2  V-5  (1  -  ef  (1  +  9e)  ~  *  £  (tan  ^  +  £  tan3  ^w^. 
But 


where  #  is  the  perihelion  distance,  in  the  present  problem  a  more  convenient 
element  than  the  mean  distance  a.     Hence 

l  +  9e\   t-U 

•  —^?r-    •  — r>-  =  tan  iw,  +  4  tan3  Aw, 
y      20    /      J5 


32  Motion  under  a  Central  Attraction  [CH.  in 

the  analogy  of  which  with  (9)  of  §  29  is  evident.  Here  B  is  unknown,  but 
the  supposition  that  B  =  1  will  lead  to  a  good  first  approximation  to  tan  ^wl 
and  hence  to  A,  and  a  nearer  value  for  log  B  can  then  be  taken  from  the  table. 
This  in  turn  will  lead  to  a  second  approximation  to  tan^wl5  and  so  on  until 
the  correct  value  is.  reached.  Now  let 

T  =  tan2    E  = 


or 


where  G  is  a  function  of  the  second  order  in  A,  i.e.  a  small  quantity  of  the 
fourth  order  in  E,  which  like  log  B  can  be  tabulated  with  the  argument  A. 
Hence 

/-       //l+e\         //1  +  e  A 

tan  iw  =  V  T  .  A  /  )  =*  A  /  1  1 

\\l-e)     V  \1  - 


e      - 


Finally,  by  §  27, 

r  cos2  fyv  -  a  (1  -  e)  cos2  J^^  =  ^/(l  +  T) 
or 


r  = 


so  that  the  problem  of  finding  w  and  r  is  solved  by  the  aid  of  the  tables 
giving  log  B  and  C  with  the  argument  A  without  introducing  E  explicitly 
into  the  calculation.  The  method  with  very  little  change  is  adapted  equally 
to  hyperbolic  orbits.  The  tables  will  be  found  in  the  Theoria  Motus  of  Gauss, 
or  in  an  equivalent  form  in  Bauschinger's  Tafeln,  Nos.  xvn  and  xvin. 


CHAPTER  IV 

EXPANSIONS   IN    ELLIPTIC    MOTION 

35.     The  fundamental   equations  of  elliptic  motion  found  in  the   last 
chapter,  namely 

M=E-esinE,     e  =  sin<£  (1) 


•(2) 


'1  +  e\ 
tan  ^ w  =  .  I  (  —    —  I  tan  \ E  =  tan  (|<£  + 1  TT)  tan 

v    VI  —  6/ 

— ~  tan  iJ^,     /3  =  tan  ^<b 


=  l-ecosE (3) 


a     1  +  e  cos  w 

give  at  once  the  means  of  calculating  the  coordinates  at  any  given  time.  But 
for  many  purposes  it  is  necessary  to  express  them  as  periodic  functions  in  the' 
form  of  series.  Some  of  the  more  important  forms  of  expansion  will  now  be 
investigated. 

But  certain  changes  in  these  equations  are  sometimes  useful.     Let 

iw  =  log  x,     iE  =  log  y,     iM  =  log  z,     i2  =  —  1. 

Then  from  (2) 

x-l _l+8  y-I 
x  +  1  ~  i—$'y  +  1 

y -  8  x  +  8 

/-»»  ——       Si  '}!    • — • , 

Also  by  (1) 


-7/-1)]  ...........................  ........................  (4) 

-8   (^ 


=__ 

3' 


-^  exp.  [8  cos  <f>  {(/8  +  #)-'  -  (8  +  x'1)-1}]  .  .  .(5) 

P.  D.  A.  3 


34  Expansions  in  Elliptic  Motion  [CH.  iv 

The  equation  (3)  gives 


/I    _   /02\2 


It  is  evident  that  some  expansions  will  be  made  more  simply  in  terms  of 
ft  than  of  e.  Hence  it  will  be  useful  to  have  the  development  of  any  positive 
power  of  ft  in  terms  of  e.  Now 

ft  +  ft~l  =  tan  $(f>  +  cot  ^<j>  =  2  cosec  <f>  =  2e-1 
or 

0  =  0  +  & 

Hence  by  Lagrange's  theorem 


" 


for  the  only  terms  which  survive  arise  when  q  =  2p  +  m.     Hence 


...      ...(7) 


and  it  is  readily  seen  that  this  series  is  absolutely  convergent. 

36.     Since 

*  =  (2/-/3)(l  -/%)- 
it  follows  that 

log  x  =  log  y  +  log  (1  -  /fy-1)  -  log  (1  - 
Hence 


.)     .........  (8) 

But  x  and  y  can  be  interchanged  if  the  sign  of  ft  is  changed  at  the  same  time. 
Therefore 


—   /    sn   w  +        sn    w  —.... 

It  is  also  easy  to  express  M  in  terms  of  w.     For,  by  (5), 
log  z  =  log  a  +  log  (1  +  ftx-1}  -  log  (1  +  ftx)  +  ft  cos  6  {(x  +  /S)-1  -  (or1  +  /3)-1} 
=  log  x  -  ft  (x  -  x~l)  +  ^ft-  (x2  -  #-2)  -  |/33  (ar5  -  x~3)  +  ... 

+  ft  cos  </>{-(>-  a:-1)  +  ft  (x-  -  x~*)  -  ft-  (XA  -  x~s)  +  ...  j 
=  log  x  -  13  (  I  +  cos  0)  (as  -  x~l)  +  /32  ($  +  cos 


35-37]  Expansions  in  Elliptic  Motion  35 

and  therefore 

M  =  w  —  2  {/3(1  +  cos  ^>)  sin  w  —  /32(£  +  cos  <£)sin  2w  +  /33(^+cos<£)sin3w— ...}. 

By  this  expansion  the  equation  of  the  centre,  w  —  M,  is  expressed  as  a  series  in 
terms  of  the  true  anomaly. 

37.  We  have  now  to  consider  the  expansions  in  terms  of  M~,  which  are  of 
the  greatest  importance  because  they  are  required  in  order  to  express  the 
coordinates  as  periodic  functions  of  the  time.  And  first  we  take  the  case  of 
r"1.  Now 

a  dE 

-  =  (1  — ecosA)  *  =  -pjri . 
r  dm 

This  is  an  even  periodic  function  of  E  and  consequently  of  M.     Hence 

a      1  'w  2  f71" 

(l-ecos  E)-1  dM  +  2  -  cos  pM      (l-e  cos  E)~l  cos pMdM 

>'         If.  0  7T  Jo 

2  frr 

—  ScosjxMI  cos(pE  —  pesinE)dE 

it  J  o 

00 

=  1  +  2  2J  «7j,  (^e)  cospM (9) 

where 

1  [* 

J  (pe)  =  —     cos  ( pE  —  pe  sin  E )  dE. 

TTJo 

Jp  (pe)  is  called  the  Bessel's  coefficient  of  order  p  and  argument  |?e.  We  shall 
briefly  study  the  properties  of  these  coefficients  so  far  as  they  are  required  for 
our  immediate  purpose. 

Let 

F(t)  =  exp.  [\x  (t  -  r1)}  =  2apV>. 

—  CO 

For  t  write  exp.  (—  ti/r).     Then 

•4-  <x> 

exp.  (—  ix  sin  ty)  =  "2,  ap  exp.  (-  ipty). 
This  is  a  Fourier  expansion,  showing  that 

ap  =  ^—       exp.  i  (p-^r  —  x  sin  -\Jr)  rfi|r 


and  combining  the  parts  of  the  integral  which  are  due  to  i/r  and  2-Tr  —  i/r  we 
have 


1 

a  —-     Cos    ?-»r  —  a;  sn 


3—2 


36  Expansions  in  Elliptic  Motion  [CH.  iv 

Thus  the  coefficients  in  the  expansion  of  F(t)  are  precisely  the  coefficients 
which  we  have  to  study.     Now 

F(t)  =  exp.  (%xt)  exp.  (— 


Hence  Jp(x}  is  the  coefficient  of  those  terms  for  which  a  =  @+p,  or 


If  p  is  positive,  £  takes  the  values  0,  1,  2,...  and  the  expansion  becomes 

\_     xP     (i 
p(a    = 


If  j9  is  negative,  /3  takes  the  values  —  p,  —  p  +  1,  .  .  .  ,  because  a  cannot  be  negative. 

38.     The  effect  of  changing  the  signs  of  x  and  t  is  to  leave  F(t)  unaltered. 
HGHCG 

jp(x}  =  (-iyjp(-x)    .......................  (12) 

Similarly  F(t)  is  unchanged  if  —  1~*  is  substituted  for  t.     Hence 

Jp(u)  =  (-VrJ^(x)   ...........................  (13) 

Again,  the  result  of  differentiating  F(t)  with  respect  to  t,  gives 

Ja?  (1  +  t~z\  2  Jp  (x)  tP  =  2pJp  (^  tP~\ 
Equating  the  coefficients  of  tp~l  we  have 

&{Jp-i(a)+Jp+i(*j\=pJp(ab  .....................  (14) 

On  the  other  hand,  if  we  differentiate  F(t)  with  respect  to  x,  we  have 


or,  equating  the  coefficients  of  tp, 

%{Jp^(x)-Jp+l(x)}=Jp'(x)  .....................  (15) 

These  simple  recurrence  formulae  show  that,  with  any  given  argument,  Bessel's 
coefficients  of  any  order,  and  their  derivatives,  can  be  expressed  as  linear 
functions  of  the  coefficients  of  any  two  particular  orders,  or  of  any  one 
coefficient  and  its  derivative,  e.g.  </,  (x)  and  //(#).  In  particular, 


=  i  {Jp_2  (x)  -  2,/p  O)  +  Jp+2  (x)} 
=  -Jp  Or)  +  —  {(p  -  1)  Jp_,  (x)  +  (p 


37-39]  Expansions  in  Elliptic  Motion  37 

or 


This  shows  that  JP{x)  is  a  particular  solution  of  the  equation 

d2y      1  dy 


The  general  theory  of  Bessel's  functions,  denned  as  solutions  of  this  dif- 
ferential equation,  is  not  required  for  our  purpose.  We  need  only  the 
solutions  of  the  first  kind,  with  integral  values  of  p,  and  the  definition  given 
above  is  sufficient. 

39.     The   desired   expansions   in   M   can   now   be   resumed.     We   take 
sin  mE  which  is  an  odd  function  of  E  and  M.     Therefore 


2  f77 

sin  mE  =      -  2  sin  pM  \    sin  mE  sin  pM  dM 


2  ^  ,  f  "•  1 


= 2  sin  pM  I     -  sin  mE .  d  {cos  (pE  —  pe  sin  E)} 


2  i**"  w 

=      -  2  sin  pM       —  cos  mE  cos  ( »-£"  —  pe  sin  £")  d.E' 
TT  J0  P 


fff  TfL  

{cos(«  -mE-pesin  E) 
Jo    P 


2 

2,  sm  73^  I 

!o  P 

(by  integration  by  parts,  the  integrated  part  vanishing  at  the  limits) 
1^ 

7T 


4-  cos  (p  +  mE  —pe  sin  E)}  dE 

^  sin  pM  ,  r 

^~~{Jp_m(pe)  +  Jp+m(pe)}    ...............  (17) 


In  particular,  when  m  =  1,  by  (14) 

.     „     2  ^  sin  p  M    , 
s™E=-2—-?—.Jp(Pe)  .....................  (18) 

and  therefore 

...............  (19) 


Similarly,  since  cos  mE  is  an  even  function  of  E  and  M, 

2  i""" 

cos  mE  =  a0  H  —  2  cos  pM     cos  mE  cos  pM  dM 

7T  J0 

2  /"""  1 

=  a0  +  —  2  cos  «./¥      -  cos<mE  .  d  !sin  (  pE  —  pe  sin  E}\ 
TT  J0  J? 

2  r™1  m 

=  a0  +  -  2  cospM       —  sin  raJf?  sin  (».&  —  »e  sin  £")  c?£" 

7T  Jo    /> 


38  Expansions  in  Elliptic  Motion  [CH.  iv 

(integrating  by  parts  as  before) 

.  =  a0  +  -  S  cos  pM       -  {cos  (p  —  mE  —pe  sin  E} 

' 


—  cos  (p  +  mE  —pe  sin  E)}  dE 
—  {Jp-m(pe)-Jp+m(pe)}  ..................  (20) 

The  constant  term  has  not  been  determined.     It  is 

1   [* 

a0  =  —I   cos  mE  dM 
wJo 

1   f" 
=  -      cos  m£"  (1  -  e  cos  J£)  dE 

7T  Jo 
1     f  "" 

=  -      {cos  mE  -  \e  cos  (m  +  1)  2?  —  \e  cos  (w  —  1)  .#}  d# 

T  Jo 

and  thus 

a0  =      1    if  m  =  0 

=  —  \e  if  m=  1 
=      0.  if  m>l. 
The  particular  case  of  m  =  1  is  simplified  by  (15),  so  that 

..................  (21) 


40.     From  the  last  expansion  it  follows  that 

.........  (22) 


Any  positive  power  of  r  can  be  expanded  by  means  of  (20).     For  example 

7.2 

-  =  (1  -  e  cos  E)- 

\Ju 

=  l+^-2e  cos  E  +  |e2  cos  2E 

_,  cos  pM  T  .  „  ,,  cos  pM  ,  , 

=  1  +  |e2  +  e2  -  4,e  2  —  Z—  Jp'  (pe)  +  ez^  —  *—  [Jp^  (pe)  -Jp+2  (pe)}. 

Now,  by  (14)  and  (15), 

2(jt>-l)  ,  2(p  +  l)  T 

JP-*  (pe)  -  Jp+*  (pe)  =  -  J  Jp-,  (pe)  -  -  Jp+l  (pe) 

•          _/^  f^ 

4  4 

=  -  Jp'(pe)-~  Jp(pe). 

Hence 


(23) 

p 


39-4.1  ]  Expansions  in  Elliptic  Motion  39 

The  expansions  of  the  rectangular  coordinates  can  be  written  down  at  once 
by  means  of  (18)  and  (21).     Thus,  if  x,  y  have  this  meaning  and  not  as  in  §  35, 

x  =  a  cos  E  —  ae 

nnc  11  n/f  \ 

(24) 


(25) 


and 

—  e2}  a  sin  E 


Other  important  expansions  can  be  derived  from  those  already  obtained  by 
differentiation  or  integration.     For  instance,  the  equations  of  motion  give 

directly 

d?x      a?x 
= 


whence 

^=-,2pJp(pe)cospM  .....  .  ..................  (26) 

y      2 


(27) 


41.     The  expansion  of  functions  of  the  true  anomaly  in  terms  of  the 
mean  anomaly  is  in  general  more  difficult.     But  sin  w  and  cos  w  are  readily 

found.     For  (§  27) 

-  e2)  sin  E 


sin  w  = 


1  —  e  cos  E 
.   d  ,,  dE 


(28) 

by  (22).     And 

cos  E  —  e 

cos  lu  —  ^—  „ 

1  -  e  cos  & 

-i     1~e2  « 
e     '  r 

2(l-e2)v 
by  (9). 


40  Expansions  in  Elliptic  Motion  [OH.  iv 


1  —  e- 
»in  (w  -M)  =  e  sin  M  —  '-      -  2  Jp  (pe)  [sin  (p  +  1)  M  -  sin  (p  -  1)  M] 


Hence  also  for  the  equation  of  the  centre, 

-  e- 

—  2  Jp  (pe)  {sin 

+  V(l  -  e2)  2  Jp'  (pe)  {sin  (p  +  l)M+  sin  (p  -  1)  M} 

1  —  e2  i  * 

=  ^e+-     —  J2(2e)  +  \/(l  —  e2)J2'(2e)\sin.M+  2  a^sinp.¥...(30) 

where 


ap  =  -  -      -  { Jp_!  (p  -  1 .  e)  -  Jp+l  (p  +  l.  e)} 


e2)  {J'p-,  (p-l.e)  +  J'p+l  (p  +  l.  e\}. 

This  expansion  for  the  equation  of  the  centre  in  terms  of  the  mean 
anomaly  is  important,  although  the  coefficients  are  rather  complicated. 
Hence,  as  far  as  ea, 

sin  (w  -  M)  =  e  (2  -  fe2)  sin  M  +  fe2  sin  2M  +  $e3  sin  3M 
w  -  M  =  e  (2  -  |e2)  sin  M  +  \&  sin  2M  +  |f  e3  sin  3M 
as  can  easily  be  verified. 

*42.  For  some  purposes1  Laurent  series  in  the  exponentials  x,  y,  z  of 
§  35  are  more  convenient  than  Fourier  series  in  w,  E,  M.  Clearly 

x~l  dx  =  i  dw,     y~L  dy  =  i  dE,     z~l  dz  =  I  dM. 
Let 

S  =  a0  +  2  (ap  cospd  +  bp  sin  pd} 

=  a0  +  2  {£  (ap  -  ibp}  T*>+$  (ap  +  ibp)  T-P] 
where  log  T  =  id.     By  Fourier's  theorem 

r  2ff  r  2tr 

Trttp  =       ScospddQ,     Trip  =  I     SsinpOdQ 

Jo  Jo 

/•27T  r27r 

(Op  -  ibp)  =        ST-P  dd,     TT  (ctp  +  ibp)  =       ST?  d0. 

J  0  .'0 


7T 

Hence 

S  = 
where 


/•*?•«• 

rzn- 

Jo 

This  well-known  form,  intermediate  between  Fourier's  and  Laurent's,  is 
general  and  includes  the  case  p  =  0.  It  has  been  used  already  in  §  37. 

Formulae  have  been  found  which  make  it  possible  to  pass  from  any 
Fourier's  expansion  in  E  to  one  in  M.  The  general  result  may  be  expressed 
in  a  slightly  different  way.  For,  since  y  has  the  same  period  as  z, 

*  The  reading  of  §§  42—46  can  quite  conveniently  be  deferred  till  after  Chapter  XIII. 


41-43]  Expansions  in  Elliptic  Motion  41 

where 

=  I  "  y*z~m  dM  =  im~l  |  y?d  ($-"*) 


r2n 
=  pin'1       exp.  {ipE  -  tm  (E-e  sin  E)}  dE 

Jo 

=  ^7rpm~lJm^p  (me) 
(m  ^  0).     But  when  ra  =  0, 

•2vA0  =  r  yv  dM  =  j  *  V  0--e  cos  E)  dE 

Jo  Jo 


Hence  generally,  for  any  function  of  y, 


p  ffl=±l  p 

=  B0-  \e  (Bl  +  B^)  +    S    2pm-lBpJm,p  (me)  zm. 

m- :±1  p 

43.  There  is  another  form  of  calculation,  due  to  Cauchy,  in  which  Bessel's 
coefficients  do  not  appear  explicitly.  Let  8  be  any  periodic  function,  such 
that 

Here,  by  (4), 


rt» 

Sy-'*>  exp.  \_\pe  (y-  y~l)~\  (1-ecosE)  dE 
Jo 

! 

Jo 

Jo 


"Sy^>  (1  -$e(y  +  y~1)}  exp 


-p  dE 
Jo 
where 

(31) 


the  coefficient  Bp  of  U  expanded  in  powers  of  y±l  being  thus  identical  with 
the  coefficient  Ap  of  S  expanded  in  powers  of  z±l. 


42  Expansions  in  Elliptic  Motion  [CH.  iv 

Again, 

/•2n  (]„  .    t"2w        ,]?-p 

=  -il     8z-f-*-~dM=ip-*l    S-~-fdM 
Jo  dM  Jo       dM 

r2jr       r/iSf  r2*-      r/s 

=  -ip~l       z-P^dM=-ip-*       z-p^dE 
Jo         dM  Jo         dE 


dy 

l-Zn  1  (JQ 

Jo   P  ^  ^ 

I^V 
Jo 


where 

I      JO 

F  =  p  Ty 


the  coefficient  5^  of  F  expanded  in  powers  of  y±l  being  thus  identical  with 
the  coefficient  Ap  of  8  expanded  in  powers  of  z±l.  The  form  (32)  becomes 
illusory  when  p  =  0. 

Now  the  exponential  function  occurring  in  (31),  (32)  can  be  expanded  in 
a  series  with  Bessel's  coefficients  having  the  argument  pe.  That  returns  to 
the  methods  already  considered.  But  another  process  is  possible  and  has 
advantages  if  S  is  of  suitable  form.  This  consists  in  developing  first  in 
powers  of  y  —  y~l.  Let 


p=  —oo 


where  j  and  q  are  integers  (not  negative).  The  numerical  coefficients  N  are 
called  Cauchy's  numbers  and  it  is  evident  that  a  knowledge  of  them  will  be 
required  in  this  method.  By  comparing  coefficients  of  tp  in  the  identity 

(t  +  t-y+i  (t  -  1-1)?  =  r1  (t  +  rjy  (t  -  t-i)i  +  1  (t  +  t~iy  (t  -  1-^ 

it  is  evident  that 

N  -  AT"  4-  N 

•iv—  P>  J+ii  1  ~  a—f—liJi  9  T       —P+i,J,  1' 

From  a  double-entry  table  giving  N_p>  0>  q  with  the  arguments  p,  q,  therefore, 
similar  tables  giving  N_ptltg,  N_p>^q,  ...  can  be  readily  constructed.  The 
effect  of  interchanging  t  and  t~l  shows  that 


The  expansion  is  either  even  or  odd  and  the  highest  term  is  V+q.     Hence 
j  +  q  —  p  is  a  positive  even  integer,  and  if  p  =j  +  q,  N=l. 


43,  44]  Expansions  in  Elliptic  Motion  43 

It  is  now  only  necessary  to  consider  the  construction  of  the  table  for 
W-P,  o,  q  when  p  is  positive.     But  this  is  indicated  by 

(t  -  t-y  =  2,  N_p,  „,  g  «*  =  2    .7q  !-T-.  tr  (-  ry-r 

rl(q  —  r)l 
whence  p  =  2r  —  q,  and 

a  * 
W  —  (—~\  ^i<?-p)  _  ^  ' 

[*(?+/>)]  iftte-jW- 

The  tabulation  of  Cauchy's  numbers,   which  are  all   positive  or  negative 
integers,  is  therefore  an  extremely  simple  matter. 

44.     To  consider  an  example,  let 

S  =  (J  -  1  Y"  =  (-e  cos  E)m  =  (-  \e)m  (y  +  y~l)m. 
Then 

U  =  {(-  \e)m  (y  +  y~l)m  +  (~  i*)m+1  (V  +  2/"1)m+1l  exp.  \_\pe  (y  -  y-1)] 

=  {(-  &)m  (y  +  y~T  +  (-  ^)m+l  (y  +  rl)m+l}  2  (W1  (y  -  y~l}ql<i  ' 
-  (-  W  (y  +  y~iyn  2  (ipey  (y  -  y-^fg  i 

q 

+  (-.i*)-*1  (y  +  y~l)m+l  2  (IpO*-1  (y  -  y~l)q-ll(g  -  1)  1 
« 

and 

B  -  /_  ifl\»  v  <i£!>!  [jv        -?AT 

^  —  V       2e^      *      Tl  1V~P,  >»,  9        ~  IV-P,  *»+!,  9-1 

</        ?  •        L  P  J 

is  the  coefficient  of  y*  in,  U,  and  therefore  of  0^  in  S. 

When  p  =  0  the  exponential  function  disappears  and  the  constant  term  is 
given  by 

U=(-  \e)m  (y  +  y~l)m  +  (-  le)m+l  (y  +  y-i)m+» 

and  is  therefore  the  first  or  the  second  of  the  forms 


(le)m  m  I  [(|m)  !]-2,     (le)'^1  (ro  +  1)  !{&(*»  +  1)]  !j 
according  as  m  is  even  or  odd. 
On  the  other  hand, 


and  therefore 

F  =  ^  (-  ^)»  jr»  (y 

P  q      7  : 

Hence 


. 

p-l  —         V       2  *>  ,  -p,  7)1-1, 

P  9         ^  • 


44  Expansions  in  Elliptic  Motion  [OH.  iv 

is  the  coefficient  of  yp~l  in  V  and  therefore  also  the  coefficient  of  zp  in  S. 
Comparison  with  the  previous  result  shows  that 

A7  —       A7^  A7 

//c-XV  — p^  ffi — j^  q-\-i  ~~~~  Y'-^-V — p^  j/j,(  q         1^-i.Y — p^  ?/3,-}-;i(  q — i 

is  an  identity.     From  this  the  recurrence  formula 

(m  —  p  +  q  +  2) N_p+2t m!g-2(m  —  q)  N_p< m< q  +  (m  +  p  +  q  +  2) N-p-z, m,  q 
can  be  easily  deduced. 

45.     The  development  in  terms  of  M  or  z  of  the  functions 

fr\n  sin  fr\n    0 

mw,  xm 

\aj    cos  \a/ 

is  of  special  importance.  Here  n  is  any  positive  or  negative  integer,  and  if 
m  is  also  a  positive  or  negative  integer  it  is  only  necessary  to  consider  the 
second  form.  This  involves  Hansens  coefficients  X"'  m,  where 


Now 

r  /r\2  1  -I-  /92  /r\2 

dM=-dE=(-1  secd>dw=-^~(-}  dw 
a  \aj  1  -  y8'2  \a/ 

of  which  the  last  form  follows  from  the  areal  property  of  elliptic  motion, 

r2dw  =  hdt  =  n~}hdM  =  ab  .  dM  =  a2  cos  <f>dM. 
Also 


and  therefore  X™'  m  can  be  expressed  by  a  definite  integral  involving  y  and 
E,  or  by  one  involving  x  and  w,  by  means  of  (4),  (5),  (6),  thus 


exp.  [$ie  (y  -  y~1)]  dE 
and 

O 

(I-  @2)M+S  (1  +  /3*)-»-i  ««-<  (1  +  00)-"-**  (1  +  /3^-1)-w~2-i 
o 

exp.  [i/9  cos  <^>  {(/3  +  a--1)-1  -  (y3  4-  a)"1}]  dw. 

The  first  of  these  forms  shows  that  (1.+  /32)n+1Z"'w  is  the  coefficient  of  y*'-™ 
in  the  expanded  product  F]F2,  where 

F!  =  (1  -  j3y)>l+l-m  exp.  (^tey) 

F2  =  (1  -  ^y-i^+i+m  exp.  (-  Itey-1)- 

Similarly  the  second  form   shows  that  (1+  /S8)"*1  (1  -  $*)-™-3Xn.'m  is  the 
coefficient  of  aei~m  in  the  expanded  product  X^XZ,  where 

X,  =  (1  +  /3#)-n-2+i  exp.  [i  cos  <f>  .  /3x  (1  +  /Sa?)-1] 

JT2  =  (1  +  yS^-1)-71"2^  exp.  [-  i  cos  0  .  /&;-1  (1  +  /Sar1)"1]. 


44,  45]  Expansions  in  Elliptic  Motion  45 

The  deduction  of  Hansen's  formulae  in  this  way  is  not  difficult,  and  has  been 
given  by  Tisserand  (Mec.  Gel.,  I,  ch.  xv). 

An  obvious  method  consists  in  expanding  the  exponential  function  oc- 
curring in  the  first  of  the  two  integral  forms  in  a  series  with  Bessel's 
coefficients.  Thus 

'"  =  (1  +  fl2)-"-1  2  Jv  (ie)  I  *  y>+™~*  (1  -  /3y)n+1~m  (1  -^u-l)n+1+m  dE 

' 


p 
where  Xn'm  is  clearly  the  coefficient  of  yi~P~m  in  the  expansion  of 

r1  (0)  =  (1  -  fly)**-"  (1  -  /3y~r+l+m 
and  therefore  equally  the  coefficient  of  y-i+p+m  in  the  expansion  of 

Yn_m  (£)  =  (!-  /3y-i)n+i-m  (1  -  (3y)n+l+m. 
Now 


-        , 

-  2  r-  IV  #**  VP  i-V-P-k  +  V  j- 

~ 


where  h=p  +  k,  and  if  ^  is  positive  the  coefficient  of  yp  is 

/     R\P  i-~(i-P  +  V  *  (i-p)...(i-p-k  +  \)  j...(j-k  +  l) 
p\  (p  +  l)...(p  +  k)  kl 


in  the  ordinary  notation  for  a  hypergeometric  series.     Hence  there  are  two 
possible  forms  for  X™'  ™ : 


F(i—p  —  n  —  l,  —m—n  —  l,  i  —  p—m  +  l,  /32) 
p  —  ml 

'(—  i  +  p  —  n  —  I,  m  —  n  —  l,  —i  +  p  +  m  +  l,  /32) 

of  which  the  first  is  available  if  i  —  p  —  m >  0  and  the  second  if  i—p  —  m<0, 
for  then  the  third  argument  of  the  series  is  positive  and  the  binomial  coeffi- 
cient has  a  meaning.  If  i  —  p  =  m  both  forms  become 

Xn' m  =  F (m  —  n  —  I,  —  m  —  n  -  I,  I,  /32). 

19  p 

When  n  is  assumed  to  be  positive,  at  least  one  of  the  first  two  arguments  of 
the  series  is  always  negative,  and  therefore  the  series  is  a  polynomial  in  $2. 
For  in  the  first  form  with  i  —  p  —  m  >  0,  the  second  argument  is  certainly 


46  Expansions  in  Elliptic  Motion  [CH.  iv 

negative  if  m  is  positive  ;  if  m  is  negative,  n  +  1  —  m  >  0  and  the  binomial 
coefficient  shows  that  i  —  p  —  m  <  n  +  1  —  m,  so  that  the  first  argument  is 
negative.  Similarly  when  the  second  form  is  valid  it  also  is  a  terminating 
series.  When  n  is  negative  one  of  the  known  transformations  of  the 
hypergeometric  series  may  be  necessary  to  give  a  finite  form.  Hence 
Hansen's  coefficients  are  reduced  to  the  form 


where  X"'m  represents,  with  a  simple  factor,  a  hypergeometric  polynomial 
in  /32.  This  form  was  first  given  by  Hill. 

46.  The  periodic  series  in  M  found  above  are  evidently  legitimate 
Fourier  expansions,  satisfying  the  necessary  conditions  with  e  <  1,  and  as 
such  are  convergent.  The  Bessel's  coefficients  are  given  in  explicit  form  by 
the  series  (11)  which  also  is  at  once  seen  to  be  absolutely  convergent  for 
all  values  of  e.  But  in  practical  applications  the  expansions  are  generally 
ordered  not  as  Fourier  series  in  M  but  as  power  series  in  e.  Under  these 
circumstances  the  question  of  convergence  is  altered  and  needs  a  special 
investigation.  Now 

E  =  M  +  e  sin  E 

considered  as  an  equation  in  E  has  one  root  in  the  interior  of  a  given  contour, 
and  any  regular  function  of  this  root  can  be  expanded  by  Lagrange's  theorem 
as  a  power  series  in  e,  provided  that 

\esrn  E\<\E-M 

at  all  points  of  the  given  contour*.  We  have  then  to  find  a  contour  with  the 
required  property,  and  to  examine  its  limits. 

We  are  to  regard  e  and  M  as  given  real  constants.     The  equation 

E  =  M  +  p  cos  %  +  t>p  sin  ^ 
where  p  is  constant,  defines  a  circular  contour.     At  any  point  on  it 

•  sin  E  =  sin  (M  +  p  cos  %)  cosh  (/>  sin  %)  4-  1  cos  (  M  +  p  cos  %)  sinh  (p  sin  ^) 
so  that 

|  sin  E  j2  =  sin2  (M  +  p  cos  ^)  cosh2  (p  sin  ^)  +  cos2  (  M  +  p  cos  ^)  sinh2  (p  sin  %) 
=  cosh2  (p  sin  ^)  —  cos2  (  M  +  p  cos  ^) 


while 

\E-M 


*  Cf.  Whittaker's  Modern  Analysis,  p.  106 ;  Whittaker  and  Watson,  p.  133. 


45-47  1  Expansions  in  Elliptic  Motion  47 

The  most  unfavourable  point  on  the  contour  for  the  required  condition  is 
that  at  which  [  sin  E  is  greatest.  And  our  series  is  to  be  valid  for  all  real 
values  of  M.  Hence  the  condition  is  always  fulfilled  if  it  is  fulfilled  when 

sin  %  =  +  1,     cos  (M  +  p  cos  ^)  —  0 
or 

%  =  ±  fa,       M  =  ±^7T 

in  which  case 

|  sin  E  =  cosh  p. 

Thus  the  required  condition  becomes 

e  <  p/cosh  p. 

The  greatest  value  of  e  is  therefore  limited  by  the  maximum  value  of 
p/cosh  p,  which  is  given  by 

cosh  p  =  p  sinh  p, 

Inspection  of  a  table  of  hyperbolic  cosines  shows  at  once  that  p/cosh  p  is 
greatest  when  p  is  about  1'20  and  that  its  value  is  then  about  f.  With 
ordinary  logarithmic  tables  an  accurate  value  can  be  obtained  without 
difficulty  thus.  Let  tan  a  be  the  greatest  possible  value  of  e,  so  that 

tan  a  =  p/cosh.  p  =  1/sinh  p. 
It  easily  follows  that 

exp.  p  =  cot  \OL,     coth  p  =  sec  a 

whence,  by  the  equation  giving  p, 

cos  a  Log  cot  |  a  =  1 
or,  using  common  logarithms  and  taking  logarithms  once  more, 

log  cos  a  +  log  log  coti  a  +  0*362  21569  =  0. 
In  this  form  it  is  easily  verified  that 

a  =  33°  32'  3"-0,     tan  a  =  0'662  7434  .  .  .  . 

This  last  number  is  then  the  limiting  value  of  e,  within  which  the  expansion 
of  any  regular  function  of  E  in  powers  of  e  is  valid  for  all  values  of  M.  The 
orbits  of  the  members  of  the  solar  system  have  eccentricities  which  are  much 
below  this  limit,  with  the  exception  of  some,  but  not  all,  of  the  periodic 
comets. 

47.  In  the  form  in  which  Bessel's  coefficients  occur  most  frequently  in 
astronomical  expansions, 


2  .  4  .  (2j  +  2)  (2j~+  4) 


48  Expansions  in  Elliptic  Motion  [CH.  iv 

It  may  be  convenient  for  reference  to  give  the  following  table : 

e4          es 


8      192     9216 

-J2(2e)  = 
e 

&    f     .  _    .          t/G 


e  8   V         16      640 

2  _  4e3  /        4e2     4e4 

eJ4(4       =_^i    ._  +  -_ 


2  ,        ^_625e4/        25e2      625e4 
Js^       ="~ 


-\     T"   /    /      \  ~t  *^^  ^^  '  ^ 

SJ''W   '        •  ~8  +  192-  9216 

(Op2          P4  ,,6 

1-T+8-f0  + 


625e4  /,       35e2     375e4 


These  can  easily  be  carried  further  if  necessary,  but  they  are  often  enough  for 
practical  purposes. 

Bessel's  Coefficients  occur  naturally  in  several  physical  problems  discussed 
by  Euler  and  D.  Bernoulli  from  1732  onwards.  In  1771  Lagrange*  gave 
the  expression  of  the  eccentric  anomaly  in  terms  of  the  mean  anomaly,  the 
result  (19)  above,  and  found  the  expansions  of  the  coefficients  as  power  series, 
thus  anticipating  Bessel's  work  (1824)  of  more  than  half  a  century  later. 

*  Oeuvres,  in,   p.  130.     This  reference,  which  seems  to  have   been   overlooked,  is  due  to 
Prof.  Whittaker. 


RELATIONS    BETWEEN    TWO    OR    MORE    POSITIONS    IN    AN    ORBIT 

AND    THE    TIME 

48.  Since  a  conic  section  can  be  chosen  to  satisfy  any  five  conditions  it  is 
evident  that  when  the  focus  is  given,  and  two  points  on  the  curve,  an  infinite 
number  of  orbits  will  pass  through  them.  The  orbit  becomes  determinate 
when  the  length  of  the  transverse  axis  is  given,  though  in  general  the  solution 
is  not  unique.  For  let  the  points  be  P1;  P2  and  the  focal  distances  r1}  r2. 
In  the  first  place  we  take  an  elliptic  orbit  with  major  axis  2a.  The  second 
focus  lies  on  the  circle  with  centre  Pj  and  radius  2a  —  r^ ;  it  also  lies  on  the 
circle  with  radius  P2  and  radius  2a  —r.2.  These  two  circles  intersect  in  two 
points  provided  (c  being  the  length  of  the  chord  PjPg) 

2a  —  7-j  +  2a  —  i\  >  c 
or 

4a>  7*1  +  »-2  +  c (1) 

If  this  inequality  be  satisfied  two  orbits  fulfil  the  given  conditions;  if  not, 
no  such  orbit  exists.  We  notice  that  the  two  intersections  lie  on  opposite 
sides  of  the  chord  PiP2,  so  that  in  the  one  case  the  two  foci  lie  on  the  same 
side  of  the  chord,  in  the  other  on  opposite  sides.  In  other  words,  in  one 
orbit  the  chord  intersects  the  axis  at  some  point  between  the  foci,  while 
in  the  other  orbit  it  does  not.  Only  when  4>a  =  r1  +  r.2  +  c  the  two  circles 
mentioned  touch  one  another  in  a  single  point  on  PjP2  and  the  two  orbits 
coincide.  In  this  case  the  chord  passes  through  the  second  focus. 

When  the  orbit  is  the  concave  branch  of  an  hyperbola  the  second  focus 
lies  on  the  circle  with  centre  Pj  and  radius  ^  +  2«  and  also  on  the  circle 
with  centre  P2  and  radius  r2  +  2a.  These  circles  always  intersect  in  two 
distinct  real  points  since 

rl  +  2a  +  r2  +  2a>  c 

always.     There  are  therefore  always  two  hyperbolas  which  satisfy  the  con- 
ditions.    The  second  foci  lie  on  opposite  sides  of  the  chord  and  hence  in  the 
one  case  the  chord  intersects  the  axis  between  the  two  foci  and  the  difference 
p.  D.  A.  4 


50  Relations  between  two  or  more  Positions         [CH.  v 

between  the  true  anomalies  at  the  points  P15  P2  is  less  than  180°,  while  in 
the  other  case  the  chord  intersects  the  axis  beyond  the  attracting  focus  and 
the  difference  between  the  anomalies  is  greater  than  180°. 

Under  a  repulsive  force  varying  inversely  as  the  square  of  the  distance  the 
convex  branch  of  an  hyperbola  can  be  described.  The  position  of  the  second 
focus  is  again  given  by  the  intersection  of  two  circles,  the  one  with  centre  Pj 
and  radius  i\  —  2a  and  the  other  with  centre  P2  and  radius  r2  —  2a.  These 
circles  intersect  in  two  points  provided 

i\  —  2a  +  r2  —  2a  >  c 
or 

+  r2-c  .................................  (2) 


There  are  then  two  hyperbolas  and  in  the  one  case  the  chord  intersects  the 
axis  at  a  point  between  the  two  foci  while  in  the  other  it  cuts  the  axis  at  a 
point  beyond  the  second  focus. 

*  It  is  easy  to  see  similarly  that  it  is  always  possible  to  draw  four  hyper- 
bolas such  that  one  branch  passes  through  Pj  while  the  other  branch  passes 
through  P2.  These  have  no  interest  from  the  kinematical  point  of  view 
since  it  is  impossible  for  a  particle  to  pass  from  one  branch  to  the  other. 

The  case  of  parabolic  solutions,  two  of  which  always  exist,  can  be  inferred 
from  the  foregoing  by  the  principle  of  continuity.  But  it  is  otherwise  clear 
that  the  directrix  touches  the  circles  with  centres  PI,  P2  and  radii  r1}  r2.  These 
circles,  which  intersect  in  the  focus,  have  two  real  common  tangents  either  of 
which  may  be  the  directrix.  The  corresponding  axes  are  the  perpendiculars 
from  the  focus  to  these  tangents.  In  the  case  of  the  nearer  tangent  it  is 
evident  that  the  part  of  the  axis  beyond  the  focus  intersects  the  chord  PjP2 
and  the  difference  of  the  anomalies  is  greater  than  180°.  In  the  case  of  the 
opposite  tangent,  on  the  other  hand,  it  is  the  part  of  the  axis  towards  the 
directrix  which  cuts  the  chord  and  the  difference  of  the  anomalies  is  less 
than  180°. 

These  simple  geometrical  considerations  show  that,  when  the  transverse 
axis  is  given,  two  points  on  an  orbit  may  be  joined  in  general  by  four  elliptic 
arcs  (of  two  ellipses),  by  two  concave  hyperbolic  arcs,  by  two  convex  hyper- 
bolic arcs  ;  and  in  particular  by  two  parabolic  arcs.  This  conclusion  is  qualified 
by  the  conditions  (1)  and  (2)  which  of  course  cannot  be  satisfied  simul- 
taneously. All  these  different  cases  must  present  themselves  when  we  seek 
the  time  occupied  in  passing  from  one  given  point  to  another,  as  we  shall 
at  once  see. 

49.  Let  Elf  Ez  be  the  eccentric  anomalies  at  two  points  P1}  P2  on  an 
ellipse,  and  let 

2G=E2+Ely     Zg=E,-E,. 
Then 

?"i  =  a  (1  —  e  cos  E}),     r2  =  a  (1  —  e  cos  E2) 


48-50]  in  an  Orbit  and  the  Time  51 

and 


ri  +  r2  =  2a{l-e  cos  \  (Ea  +  EJ  c 

=  2a  (1  —  e  cos  6?  cos  #). 
Again,  c  being  the  chord  P1P2, 

c2  =  a2  (cos  #2  -  cos  Etf  +  a2  (1  -  e2)  (sin  jB"2  -  sin  Etf 
=  4a2  sin2  G  sin2g  +  4aa  (1  -  e2)  cos2  6r  sin2#. 

Hence  if  we  put 

cos  h  =  e  cos  G 
then 

c2  =  4a2  sin2  g  (1  -  cos2  h) 
or 

c  =  2a  sin  #  sin  h 
and 

r-j  +  r2  =  2a  (1  —  cos  g  cos  A). 
If  further  we  now  put 

e  =  h  +  g,     8  =  h  —  g 
or 

e-8=E2-El,     coslt(e+8)  =  ecos±(E2  +  E1)....:  .......  (3) 

we  have 

n  +  r.2  +  c  =  2<z  {1  -  cos  (h  +  g)}  =  4a  sin2  ^e  ...............  (4) 

rl  +  r2  —  c  =  2a{l  -  cos  (A  —  g)}  =  4asin2iS  ...............  (5) 

But  on  the  other  hand,  if  Ez  >  El  and 
//.  =  &*(!  +m)  =  w2tt3 

the  time  £  of  describing  the  arc  PiP2  is  given  by 
nt  =  E2  -  E1  -  e  (sin  E2  —  sin  #,) 
=  e  -  8  -  2  sin  £  (e  -  8)  cos  £  (e  +  8) 
=  (e  —  8)  —  (sin  e  —  sin  8)  .................................  (6) 

where  e  and  B  are  given  by  (4)  and  (5)  in  terms  of  ^  +  r.2,  c  and  a  ;  and  this 
is  Lambert's  theorem  for  elliptic  motion. 

50.  It  is  evident  that  (4)  and  (5)  do  not  give  e  and  8  without  ambiguity, 
and  this  point  must  be  examined.  We  suppose  always  that  E2  —  El  <  360°, 
i.e.  that  the  arc  described  is  less  than  a  single  circuit  of  the  orbit  ;  and  we 
assume  that  the  eccentric  anomaly  is  reckoned  from  the  pericentre  in  the 
direction  of  motion.  Now  it  is  consistent  with  (3)  to  take  £  (e  +  8)  between 
0  and  TT  and  we  also  have  £  (e  —  8)  between  the  same  limits.  Hence  \e  lies 
between  0  and  TT  and  ^8  lies  between  —  \TT  and  +  \ir.  But  the  equation  of 
the  chord  P1P2  referred  to  the  centre  of  the  ellipse  shows  that  it  cuts  the 
axis  of  x  in  the  point 

x  =  a  cos  £  (E2  -  E^/cos  $  (E*  +  EJ,     y  =  Q 

4—2 


52  Relations  between  tivo  or  more  Positions         [CH.  v 

so  that,  if  Q  is  this  point,  A  the  pericentre  and  F1F2  the  foci, 

x  —  ae  _      cos  I  (e  —8)-  cos  J,  (e  +  8)         '   sin  |e sin  |8 
aT^tt  ~  cos  £  (E.2  -  Ej  -  cos  \  (Ez  +  &\)  ~  sin  %Ei  sin  i#2 

x  +  ae  _       cos  £  (e  —  8)  +  cos  £  (e  +  8)  cos  |e  cos  |8 

AQ  ~  x  —  a  ~  cos  £  (E2  —  L\)  -  cos  \  (JE.,  +  A^  T  sin  \Ey  sin  i  A', ' 

Now  sin  |e  and  cos  ^8  are  always  positive.  We  may  also  take  E1  less  than 
2-Tr  and  sin  ^El  positive ;  then  sin  \EZ  is  negative  or  positive  according  as 
the  arc  includes  or  does  not  include  the  pericentre.  In  the  first  equation 
the  left-hand  side  is  negative  when  the  chord  intersects  the  axis  between 
the  pericentre  and  the  first  (attracting)  focus ;  in  the  second  when  the 
intersection  falls  between  the  pericentre  and  the  second  focus.  Otherwise 
both  members  are  positive.  Hence  we  see  that  sin  |8  is  positive  if  (1)  the 
arc  contains  the  pericentre  and  the  chord  intersects  F^,  or  (2)  the  arc  does 
not  contain  the  pericentre  and  the  chord  does  not  intersect  F^A  ;  and  that 
cos  |e  is  positive  if  (3)  the  arc  contains  the  pericentre  and  the  chord  inter- 
sects FZA,  or  (4)  the  arc  does  not  contain  the  pericentre  and  the  chord  does 
not  intersect  F2A.  In  other  words,  sin  ^8  is  positive  when  the  segment 
formed  by  the  arc  and  the  chord  does  not  contain  the  first  focus,  and  cos  |e 
is  positive  when  the  segment  does  not  contain  the  second  focus. 

Let  ej  and  Sj  be  the  smallest  positive  angles  which  satisfy  (4)  and  (5). 
The  other  possible  values  are  2?r  —  ej  and  —  Sj .     If  we  put 

ntz  =  e1  —  sin  ej ,     nt^  =  Si  —  sin  8^ 

• 

there  are  four  cases  to  be  distinguished,  namely: 

(a)  t  =  t3-tl 

when  the  segment  contains  neither  focus; 

(6)  ,  t  =  t.2  + 1! 

when  the  segment  contains  the  attracting,  but  not  the  other  focus ; 

(c)  t  =  2-rr/n  -L-t, 

when  the  segment  contains  the  second,  but  not  the  attracting  focus; 

(d)  t  =  27T/M  -ti  +  t 

when  the  segment  contains  both  foci.  It  is  easy  to  see  from  §  48  that  when 
the  extreme  points  of  the  arc  alone  are.  given  these  four  cases  are  always 
presented  by  the  geometrical  conditions  and  can  only  be  distinguished  by 
further  knowledge  of  the  circumstances.  Usually  it  is  known  that  the  arc  is 
comparatively  short  and  hence  that  the  solution  (a)  is  the  right  one. 


50-52]  in  an  Orbit  and  the  Time  53 

51.  The  corresponding  theorem  for  parabolic  motion  is  easily  deduced  as 
a  limiting  case.  For  when  a  is  very  large  e  and  S  are  very  small.  Hence 
(4)  and  (5)  become 

ae-  =  T!  +  rz  -f  c,     «S2  =  i\  +  n  —  c. 

1  3 

At  the  same  time,  if  we  replace  n  by  /*  /a  ,  (6)  becomes 


=  iO'i  +  r2  +  c)f  +  $  (r,  +  r2-cf. 

As  this  applies  to  the  motion  of  a  comet,  and  the  mass  of  a  comet  may  be 
considered  negligible,  we  may  therefore  write 

6kt  =  fc  +  r2  +  cf  +  (r,  +  r2  -  cf  .....................  (7) 

which  is  the  required  equation.  It  was  first  found  by  Euler.  As  regards 
the  ambiguous  sign,  the  second  focus  is  at  an  infinite  distance  and  does  not 
come  into  consideration.  But  8  is  negative  or  positive  according  as  the,. 
segment  formed  by  the  arc  described  and  the  chord  contains  or  does  not 
contain  the  focus  of  the  parabola.  Hence  the  lower  (+)  sign  is  to  be  used 
when  the  angle  described  by  the  radius  vector  exceeds  180°,  and  the  upper 
(—  )  sign  is  to  be  used  when  this  angle  is  less  than  180°,  as  it  almost 
always  is  in  actual  problems. 

52.     The  solution  of  (7)  as  an  equation  in  c  is  facilitated  by  a  trans- 
formation due  to  Encke.     We  put 

c  =  (n  +  r2)  sin  7,     0  <  7  <  90° 
and 

r)  =  2fo/(r1  +  r2)f  . 
Then  (7)  becomes 

3?7  =  (1  +  sin  7)f  +  (1  -  sin  7)* 

=  (cos  £7  +  sin  |7)3  +  (cos  £7  —  sin  %y)3  ...............  (8) 

First  we  take  the  upper  sign,  in  which  case 

877  =  6  sin  1  7  cos2  £7+2  sin3  ^7 

=  6  sin  ^7  —  4  sin3  ^7. 
If  we  put 

sin  fy  =  \/2  sin  £0,     0  <  J@  <  30° 
then 

377  =  2V2sin@,    0<    @  <  90°  ..............................  (9) 

and 

sin  7  =  2  V2  sin  £0  V(cos  §0). 
Hence 

c  =(rl  +  ra)vifj,   .............................................  (10) 

where  • 

/JL  =  siny/rj  =  3sin  J®  V(cos  §0)/sin®  ..................  (11) 


54  Relations  between  two  or  more  Positions         [OH.  v 

Since  /A  and  77  are  both  functions  of  (a),  fj,  can  be  tabulated  with  the  argument  77. 
When  such  a  table  is  available  (cf.  Bauschinger's  Tafeln,  No.  xxu)  and  77  is 
known,  c  is  immediately  given  by  (10). 

In  the  second  place  we  take  the  lower  sign  in  (8),  so  that 
877  =  2  cos3  £7  +  6  sin2  ^7  cos  ^7 

=  6  cos  £7  —  4  cos3  £7. 
If  now  we  put 

cos  £7  =  \/2  sin  J0,     30°  <  J©  <  45° 
then 

377  =  2V2sin®,    90°  <    ©<135°  ....................  .......  (12) 

and 

sin  7  =  2  \/2  sin  J©  V(cos  §©) 

as  before.  Hence  (10)  and  (11)  apply  equally  to  this  case,  with  the  difference 
'that  @  as  given  by  (12)  is  an  angle  in  the  second  quadrant  instead  of  the 
first.  Except  for  this  the  solution  is  formally  the  same  in  both  cases,  but 
different  tables  would  be  necessary.  The  case  of  angular  motion  exceeding 
180°,  however,  seldom  demands  consideration  in  practice. 

53.     For  motion  along  the  concave  branch  of  an  hyperbola  under  attraction 
to  the  focus  we'  have  (§  30) 

r-i  =  a  (e  cosh  El  —  1),     rz  =  a(e  cosh  E2  —  1) 
and  we  may  suppose  E2  >  E^.     Hence 

r,  +  r2  =  2a  {e  cosh  £  (E2  -  E,}  cosh  \  (E2  +  E,)  -  1  } 

=  2a  {cosh  £  (e  -  B)  cosh  $  (e  +  B)  -  1} 
where 

E1)    ......  (13) 


Again,  the  chord  c  is  given  by 

c2  =  a?  (cosh  E2  -  cosh  E^f  +  a?  (e>  -  1)  (sinh  E2  -  sinh  E,)2 
=  4a2  sinh2  £  (E2  -  E,)  sinh2  \  (E2  +  E,} 

+  4a2  (e2  -  1)  sinh2  \  (E2  -  E1}  cosh2  £  (E2  +  EJ 
=  4a2  sinh2  1  (e  -  5)  {-  1  +  cosh2  £  (e  +  5)} 
or 

c  =  2a  sinh  |  (e  -  8)  sinh  £  (e  +  3). 

Hence 

r,  +  r2  +  c  =  2a  (cosh  e  -  1)  =  4a  sinh2  £e  ...............  (14) 

t 
r1  +  r2-c=2a(coshS-  l)  =  4asinh2£S  ...............  (15) 


52-54]  in  an  Orbit  and  the  Time  55 

But  on  the  other  hand  if 

/j,  =  &2(1  +  m)  =  ri>as 

nt  =  e  sinh  E2  —  E2  —  (e  sinh  El  —  E^) 

=  2e  sinh  £  (E2  -  E\)  cosh  \  (E2  +  E,)  -  (E2  -  EJ 

=  2  sinh  1  (e  -  8)  cosh  £  (e  +  8)  -  (e  -  8) 

=  sinh  e  —  sinh  8  —  (e  —  8)  ........................  77..  7  .......  (16) 

where  e  and  8  are  given  by  (14)  and  (15).  This  is  the  form  which  Lambert's 
theorem  takes  in  this  case. 

We  may  take  ^  (e  +  8)  as  defined  by  (13)  positive  ;  and  £  (e  —  8)  is  positive 
since  A'2  >  Et.  Hence  e  is  positive.  Now  the  equation  of  the  chord  referred 
to  the  centre  of  the  hyperbola  gives  for  the  intercept  on  the  axis 

x  =  -  a  cosh  |  (E2  -  #,)/cosh  £  (E2  +  E1),    y  =  0 
or,  (—  ae,  0)  being  the  attracting  focus  within  this  branch, 

ac+ae=-a  {cosh  £  (e  -  8)  -  cosh  £  (e  +  8)}/cosh  £  (E2  +  EJ 

=  +  2a  sinh  £e  sinh  £8/cosh£  (#2  +  ^1)  .....................  (17) 

The  left-hand  side  is  negative  or  positive  according  as  the  intersection  falls 
beyond  the  focus  or  on  the  side  of  the  focus  towards  the  centre.  Hence 
sinh  ^8  is  positive  when  the  angular  motion  about  the  focus  is  less  than  180°, 
and  negative  when  it  exceeds  180°.  Thus  the  sign  of  8  is  determined.  If 
we  put 

mi2  =  fa  +  r2+  c)/4a,     m2*  =  fa  +  r2-  c)/4a 
then 

sinh  |e  =  +  ml}     sinh  ^8  =  ±  m2 
or 


_ 
exp.  |  e  =  +  ?>*!  +  \/m^  +  1,     exp.  %8  =  +  m2  +  Vm22  +  1 

sinh    e  =        2mj  Vw/  +  1,     sinh    8  =  +  2?%  VW  +  1- 
Hence  (16)  can  be  written  (Log  denoting  natural  logarithm) 


nt  =  2ml    ra^  +  1  +  2m2     m22  +  1 

-  2  Log  (TO!  +  Vmj2  +  1)  ±  2  Log  (m2  +  Vw22  +1) 

where  the  upper  or  the  lower  sign  is  to  be  taken  according  as  the  angular 
motion  about  the  attracting  focus  is  less  or  greater  than  180°.  ' 

54.  The  corresponding  theorem  for  motion  along  the  convex  branch  of 
an  hyperbola  under  a  repulsive  force  from  the  focus  can  be  proved  similarly. 
In  this  case  (§  32) 

?*!  =  a  (e  cosh  El  -}-  1),     r.,  =  a(e  cosh  E2  +  I). 
Hence 

r,  +  r2  =  2a  (cosh  \  (e  +  8)  cosh  \  (e  -  8)  +  1} 


56  Relations  between  two  or  more  Positions          [CH.  v 

where 


e-S  =  E2-L\,     cosh  £  (e  +  8)  =  e  cosh  £  (^2  +  #,)  .........  (18) 

and  as  in  §  53 

o  =  2a  sinh  \  (e  —  B)  sinh  ^  (e  +  8). 
We  have  therefore 

r!  +  r2  +  c  =  2a(coshe  +  I)  =  4acosh2|e  ...............  (19) 

r,  +  r2—  c  =  2a  (cosh  8  +  1)  =  4acoshs|8     ............  (20) 

Then  by  §  32  (22),  if  //  =  w2a3, 

wi  =  e  sinh  E2  +  E2  —  (e  sinh  ^  +  ^j) 

=  2e  sinh  £  (#2  -  #,)  cosh  £  (#2  +  #,)  +  E2  -  E, 

=  2  sinh  \  (e  -  8)  cosh  £  (e  +  8)  +  e  -  8  . 

=  sinh  e  —  sinh  8  +  e  -  8    ....................................  (21) 

where  e  and  8  are  given  by  (19)  and  (20).  This  is  analogous  to  the  other 
forms  of  Lambert's  equation. 

Putting  as  before 

Wj2  =  (T-L  +  r2  +  c)/4a,     m22  =  (n  +  r2  —  c)/4a 

we  have  of  necessity 

cosh  \  e  =  +  ml  ,     cosh  ^  8  =  +  m2 

but  there  is  again  an  ambiguity  in  the  values  of  e  and  S.  Now  we  may  take 
Ez  >  El  and  \  (e  —  8)  positive  ;  and  we  may  define  ^  (e  +  8)  as  the  positive 
value  which  satisfies  (18).  Hence  e  is  positive  and  exp.  (^'e)  >  1.  To  the 
equation  (17)  now  corresponds 

x  —  ae  =  -  2a  sinh  ^e  sinh  |S/cosh  |  (E2  +  E^) 

showing  that  8  is  positive  if  the  chord  intersects  the  axis  at  a  point  on  the 
side  of  the  focus  towards  the  centre.  It  must  be  noticed  that  this  focus  is, 
as  before,  the  focus  within  the  branch  and  not  the  centre  of  force.  Hence 
exp.  ^8  >  or  <  1  according  as  the  angular  motion  about  this  focus  <  or  >  180°. 
It  follows  that 


exp.  (£e)  =  +  in!  +  Vwj2  -  1,     exp.  (|S)  =  +  w2  ±  Vm22  -  1 


sinh  e  =  2wj  Vm^  -  1,          sinh  8  =  +  2w2  Vm22  -  1 
and  hence  that 


+  2  Log  (w^  +  Vraa2  -  1)  +  2  Log  (m2  4-  Vw22  -  1) 

where  Log  denotes  natural  logarithm  and  the  upper  or  the  lower  sign  is  to  be 
taken  according  as  the  motion  about  the  internal  focus  (not  the  centre 
of  force)  is  less  or  greater  than  180°. 

In  all  cases,  whether  the  motion  is  along  a  parabola  or  either  branch  of 
an  hyperbola,  when  two  focal  distances  are  given  in  position  and  nothing 


54,  55]  in  an  Orbit  and  the  Time  57 

more  is  known  about  the  circumstances,  the  discussion  of  |  48  shows  that 
the  ambiguities  in  the  expressions  for  the  time  of  describing  the  arc  corre- 
spond to  the  distinct  solutions  of  the  geometrical  problem.  Hence  they 
cannot  be  decided  without  further  information.  In  practice,  however,  it 
rarely  happens  that  the  angular  motion  about  a  focus  exceeds  180°  and 
this  limitation,  by  which  the  upper  sign  can  be  taken,  will  be  generally 
understood. 

55.  A  quantity  of  great  importance  in  the  determination  of  orbits  is  the 
ratio,  denoted  by  y,  of  the  sector  to  the  triangle.  The  case  of  elliptic  motion 
is  taken  first.  Since  n  =  h/ab,  where  h  is  the  constant  of  areas,  twice  the 
area  of  the  sector  is,  by  (6), 

ht  =  ab  {e  —  8  —  (sin  e  —  sin  8)j. 

But  if  (#],  i/j),  (x2,  7/2)  are  the  extremities  of  the  arc,  twice  the  area  of  the 
triangle  is 

2A  =  O'j  2/2  -#22/0 

=  ab  {sin  Ez  (cos  El  —  e)  —  sin  El  (cos  Ez  -  e)}  • 

=  ab  {sin  (j£a  -  EJ  -  2e  cos  £  (E2  +  E,}  sin  £  (E2  -  &\)} 

=  ab  {sin  (e  —  8)  —  (sin  e  —  sin  8)} 

by  (3).     Hence 

=       6-8-(sine-sm8) 
J      sin  (e  —  8)  —  (sin  e  —  sin  8) 

This  expression  contains  a  implicitly  and  this  quantity  is  to  be  eliminated. 
Let  2f  be  the  angle  between  rz  and  r2  and  let  g,  h  have  the  meaning  assigned 
to  them  in  §  49.  Then 

1  6a2  sin2  ^  e  sin2  £  8  =  (r^  +  r2  +  c)  (^  +  r2  —  c) 

=  (n  +  r.tf  -  r*  -  r*  +  2r,r2  cos  2/ 

=  4^7*2  COS2/ 

whence 

2a  (cos  g  —  cos  h)  =  2  cos/  vVjr2. 
Also  by  (4)  and  (5) 

r,  +  r.2  -  2a  (sin2  |e  +  sin2  £8) 

=  2a  (1  —  cos  g  cos  A) 
and  therefore 

?*i  +  r2  —  2  cos/cos  (/  vrjra  =  2a  sin2^. 
Again,  by  (22), 

nt 

V    -    -;  --  -.  - 

sin  2<jr  —  2  sin  #  cos  h 


sin  ^r  .  2  cos/  V7%r 


58  Relations  between  two  or  more  Positions         [CH.  v 

Hence 

y2  (»*!  +  r2  —  2  cos/cos  g  V?^r2)  =  2//,£2/(2  cos/vVjra)2 (23) 

since  w2a3  =  /a.     On  the  other  hand 

e  —  S  —  sin  (e  —  S) 
sin  (e  —  8)  —  (sin  e  —  sin  8) 

2<7  —  sin  2g 
2  sin  g  (cos  </  —  cos  h) 

aC2g  —  sin  2^) 
sin  g .  2  cos/  Vr1r2 
and  therefore 

o  /        T  \               /^                  9     sin  £Q  / c\  A  \ 

r(y'-1)=-2c     .../-=;-.•      0;^,^  '-   (2 

In  the  notation  of  Gauss  we  write 


2cos/Vr1r2  (2cost/V?-1r2)3     . 

and  then  (23)  and  (24)  become 

2/2  =  w2/(Z  +  sin2i#)  ...........................  (25) 

ys-y*  =  m?(2g-sm2g)/sm3g  .....................  (26) 

The  value  of  y  is  to  be  found  by  solving  this  pair  of  equations  in  y  and  g,  the 
solution  being  performed  by  some  method  of  approximation. 

56.  The  corresponding  ratio  in  the  case  of  a  parabola  can  be  expressed 
in  several  forms.  The  simplest  can  be  derived  as  a  limiting  case  from  the 
ellipse  when  a  is  large  and  e  and  8  are  small.  For  (22)  then  gives 

e3  -  83  _  e2  +  82  +  eg 

_  (e  _  g)3  +  63~_  gs  -         36g 

But  by  §§  51,  52 

a2e2S2  =  (r,  +  r2)2  -  c2  =  (n  +  r2)2  cos2  7. 
Hence 

_  2  (rt  +  ra)  +  (n  +  r,)  cos  7 
3  (?*!  +  r2)  cos  7 


where 

c  =  (ri  +  ^2)  gin  7- 

Thus  ?/,  like  77  and  //,,  is  a  function  of  7  (or  @)  and  can  therefore  like 
be  tabulated  with  the  argument  77,  where 


77  =  2Atf/(r,  +  r2)   =  2  sin  £7  (2  +  cos  7). 
(Cf.  Bauschinger's  Tafeln,  No.  xxn  a.) 


55-57]  in  an  Orbit  and  the  Time  59 

57.     In  the  case  of  the  branch  of  an  hyperbola  concave  to  the  focus  of 
attraction,  twice  the  area  of  the  sector  is  by  (16) 

ht  =  ab  {sinh  e  —  sinh  8  —  (e  —  8)} 

since  h  =  \f(f*,p}  =  nab.     And,  if  (#,,  y^,  (xz,  yz)  are  the  extremities  of  the  arc, 
twice  the  area  of  the  focal  triangle  is 

2A  =  xzyl  -  x^yz 

=  ab  {sinh  El  (cosh  Ez  —  e)  —  sinh  Ez  (cosh  El  —  e}} 
=  ab  {sinh  ( El  -  E2)  —  e  (sinh  El  —  sinh  Ez)} 
=  ab  {sinh  e  —  sinh  8  —  sinh  (e  —  8)} 

by  (13).     Hence 

=      sinh  e  -  sinh  8 -(e- 5) 
sinh  e  —  sinh  8  -  sinh  (e  —  8)  " 

Now  we  have  by  (14)  and  (15) 

16a2  sinh2  £  e  sinh2  \  8  =  fa  +  r2)2  -  c2 

=  4/^2  cos2/ 
or 

2  cos/ Vfar2  =  2a  (cosh  A  —  cosh  #) 

where  2A  =  e  +  S,  2$r  =  e  —  8.     Also  by  addition  of  the  same  equations  (14) 
and  (15) 

t\  +  rz  =  2a  (cosh  g  cosh  h  —  1) 
and  therefore 

TI  +  r2  -  2  cos/cosh  g-'it\ra  =  2a  sinh2  ^. 
But  by  (27) 

y  =  nt/(2  sinh  (7  cosh  h  —  sinh  2_gr) 

=  a  nt/sinh  g  (2  cos/  V»v*2) 
and  therefore 

2/2 (r*!  +  r2  —  2  cos/ cosh gr  V^r^  =  2/*<2/(2  cos/Vr,^)2    (28) 

since  w2a3  =  /u.     On  the  other  hand 

sinh  (e  —  8)  —  (e  —  8) 
"  sinh  e  —  sinh  8  —  sinh  (e  —  8) 

_  sinh  2g  —  2g 

2  sinh  g  (cosh  A  —  cosh  g) 

_          a  sinh  2g  —  2g 

2  cos/  VTY^          sinh  0 
Hence 


1)  =  (2cos/V^ '    "^-^'      (29) 

As  in  the  case  of  the  ellipse  we  write 


_    _ 
2  cos/  y  r,  r2  (  2  cos  /  v  r j  r2)3 


60  Relations  between  two  or  more  Positions         [CH.  v 

and  thus  (28)  and  (29)  become 

y*  =  m*l(i  -  sinh2  %g) (30) 

2/s  _  y*  _  ma  (sinn  2#  -  2#)/sinh3#    (31) 

This  pair  of  equations  in  y  and  g  must  be  solved  by  some  process  of  approxi- 
mation so  that  the  value  of  y  may  be  found. 

58.  The  case  of  the  branch  which  is  convex  to  a  centre  of  repulsive 
force  at  the  focus  (—  ae,  0)  needs  slight  modifications.  Twice  the  area  of  the 
sector  is  by  (21) 

ht  =  06"  (sinh  e  —  sinh  8  +  e  —  8) 

while  twice  the  area  of  the  triangle  is 

2 A  =  a?,ya  -  x.2y^ 

=  ab  {sinh  Ez  (cosh  E-^  +  e)  —  sinh  El  (cosh  E2  +  e}\ 
=  ab  {sinh  (E2  -  L\)  +  2e  sinh  $  (E2  -  E^  cosh  1  (E^  +  E,)} 
=  ab  {sinh  (e  —  8)  +  sinh  e  —  sirih  8} 
by  (18).     Hence  the  ratio  of  sector  to  triangle  is 


sinh  (e  —  8)  +  sinh  e  -  sinh  8 
In  this  case  we  have  by  (19)  and  (20) 

16a2  cosh2  ^e  cosh2  \  8  =  (?*i  +  r2)2  —  c2  =  4r1»-2  cos2/ 
or 

2  cos/VT^rv  =  2a  (cosh  A  +  cosh  g) 
and 

?"i  +  r2  =  2ct  (1  +  cosh  h  cosh  </) 

where  2A  =  e  +  8,  2#  =  e  —  &.     Hence 

2  cos/ cosh  </  vVir2  —  (r,  +  r2)  =  2u  sinh2</r. 
But  (32)  may  be  written 

y  =  nt/(sinh  2g  +  2  sinh  ^  cosh  A) 


and  therefore 

y1  (2  cos/ cosh  g  V?\r2  —  ^  —  r2)  =  2/x'^2/(2  cos/ ^1  t\r^ (33) 

since  wV  =  yu'.     Also  by  (32) 

sinh  (e  —  8)  —  (e  —  8} 
"      sinh  (e  —  8)  +  sinh  e  —  sinh  8 

_  sinh  2<7  -  2g 

2  sinh  (JT  (cosh  g  +  cosh  A) 

_  a  sinh  2^  —  2g 

~  2  cos/Vr^  '       sinh  # 


57-59] 
Hence 


in  an  Orbit  and  the  Time 

n't2  sinh  2a  —  ! 


61 


(2  cos/Vr^a)3 
If  as  before  we  write 

1  +  2£  =        ri  +  ra  m2  = 

2  cosf^r^ ' 

then  (33)  and  (34)  become 


(2 


.(35) 


0 


?-2#)/sinh3# (36) 

and  these,  again,  when  solved  by  a  method  of  approximation,  give  the  value 
of  y  in  this  case  when  r, ,  r2  and  f  are  known.. 

59.  Some  useful  approximations  can  be  obtained  from  a  proposition 
which  is  easily  proved.  Let  X  be  any  regular  function  of  t..  If  we  neglect 
powers  of  t  beyond  the  fourth  order  we  may  write 

2r  =  ao  +  cHi 
X  = 

Let  X1}  X2,  X3  be  the  values  of  X  when  t  =  —  r3,  0  and  TV  Then  we  have 
three  pairs  of  equations,  obtained  by  substituting  these  values  in  the  above. 
From  these  six  equations  the  coefficients  a0,  ...,  a4  can  be  eliminated  and  the 
result  expressed  in  determinant  form  is  clearly 

Z,        1        -T,        T32  -T33  T34  =0. 

X2     1000 

X3        1  T  Tx2  T!3 

X,     0      0         2      -6r3 

AY    0      0         2          0  0 

X3     0       0          2        6r,       12T!2 

The  determinant  can  be  calculated  without  difficulty,  and  the  result  after 
dividing  by  12^  r3  (r,  +  r3)  is 

0  =        1 2X,  TJ  +  A',  T!  (Tj2  -  T!  T8  -  T32) 

1   £\   ~\7       /  \     '         ~V      /  \    /         O      i      O  9t\ 

-  12  A  2  (T!  +  T3)  -  A  2  (TJ  +  T3)  (rx2  +  dTjTg  +  T32) 

+     1    O   V  i         V"  /          O  9\ 

12A3T3  + A3T3(T3-  —  TjTs-T!2). 

If  we  put  T2  =  TI  +  TS  and  write 

12^1  =  T2T3-T12,        12^2=T1T3  +  T22,        12^3-TlT2-T32 (37) 

this  becomes 

0  =  A>,  f  1  -  ^l]  -  X,r.2(l  +  ^pj  +  Z3r3  (l  -  ^-3)  ...(38) 


62  Relations  betiveen  two  or  more  Positions        [CH.  v 

60.     Now  in  the  case  of  the  motion  of  two  bodies  in  a  plane  we  have 
x  =  —  /JLX  I  r3,     y  —  — 


Hence  substituting  x  and  y  successively  for  X  in  the  formula  just  obtained 
we  have,  to  the  fourth  order  in  the  intervals  of  time, 


.  0  =  a?lTl  (1  +  Mi/ri3)  -  #2T2  (1  -  pA2/r2*)  +  xzr.A  (1 
0  =  ylr1(l+  At^j/n3)  -  y2T2  (1  -  pAajrf)  +  yara(l  +  pAa/raf) 

The  solution  of  these  equations  in  the  ordinary  form  gives 

^i')     T8  (1  -  /*4a/r,8)  -   r3  (1  +  nAa/ra»)       . 


^22/S         ^'32/2  •''32/1  ~T  **a2/3  «"l2/2          ^22/1 

But  the  denominators  are  respectively  double  the  areas  of  the  triangles  whose 
sides  are  pairs  of  rlt  rZ)  r3.     Hence  we  have  the  formulae  of  Gibbs, 

'  '  ' 


r,  (1  +  pAJrf)      r2  (1  -  ^,/r,")      T,  (1 

where,  according  to  the  customary  notation,  tr2ra~]  denotes  double  the  area  of 
the  triangle  whose  'sides  are  r2,  rs,  and  Al}  A2,  A3  have  the  values  found 
above  (37).  This  expresses  the  ratio  of  the  triangles  correctly  to  the  third 
order  of  the  time  intervals. 

A  second  interesting  example  is  provided  if  we  take  X  =  r2.     In  this  case 
we  have  (§§  25  and  26) 


Hence  the  formula  (38)  gives 


=  ~  (T!  (T2r3  -  T!2)  +  T2  (TIT,,  4-  r22)  +  T3  (T^  -  r32)} 

=  -  (3T!T2T3   -  T!*  +  T23  -  T:i3) 
=  —   {3TJT2T.J  -f  STjT^T,  + 

=  -yLtTjTaTs/O,       ...................................................  (40) 

The  form  (40)  applies  to  an  ellipse  and  gives  the  means  of  calculating  an 
approximate  value  of  a  when  r1;  r2,  r3  are  known.  It  must  be  adapted 
to  the  hyperbola  by  changing  the  sign  of  a.  For  the  parabola  the  right-hand 
side  vanishes  and  we  have  the  relation  between  the  three  radii  vectores 


/r,.  +  A.2r2/r2  +  Aar3/r3) 
which  holds  provided  we  may  neglect  terms  of  the  fifth  order  in  the  time. 


60,  61  ] 


in  an  Orbit  and  the  Time 


63 


61.     Returning  to  the  formula^  of  Gibbs  (39),  in  which  the  denominators 
are  correct  to  the  fourth  order,  we  have 


3  rsr8] 


[rarj 


_  1 


T,[r,r,]     l-^2r2 
to  the  third  order.     But  to  the  first  order 


=  l 


r  /M  •  A» 

1  '2  ' 


Hence 


TiJ^vyl          , 
TS  [rar,] 

TaCrirJ 


-      r  -^-iTs- 

9*     I  9*    4*    I  ->*  **  i**  * 

For  the  coefficients  we  easily  find  from  (37) 

12  (4,  +  A3)  =  TlT3  +  r22  +  Tlr2  -  rs2  =  2  (r22  -  r32) 

12  (A,  +  A2)  =  TlT3  +  T22  +  T2T3  -  T!2  =  2  (T22  -  TV) 

12  (^jTl  +  ^!T3)  =  T!  (TlT2  -  T32)  +  T3  (T2T3  -  Tj2)  =  T/  +  T33 

and  therefore 

r*y»  7^  I      T]         fi'j*  ^  4*?^  ^ 


These  formulae  are  correct  to  the  third  order  and  if  the'  terms  involving 
r 2  be  omitted  they  express  the  ratios  of  the  triangles  in  terms  of  the  single 
distance  r2  to  the  second  order.  Hence  their  value  for  the  determination  of 
orbits. 


64  Relations  between  two  or  more  Positions         [CH.  v 

62.  Without  loss  of  accuracy  the  ratio?  can  be  expressed  in  terms  of  the 
two  distances  rx  and  r3  instead  of  r2  and  r2.  The  forms  found  by  Encke 
may  be  derived  thus :  we  have  to  the  first  order 


whence 

and  therefore 


7*3  —  7"j  —  To,  Tw ,       7*i  • 
1  1 


,  =  2r2  +  r.2  (T!  -  TS) 
T^T4  (TI  -  T3) 


or 


8         ,  24(r,-ri; 


-  T, 


r2s      (r:  +  r3)3        (rx  +  r..Y          T, 
In  the  terms  of  the  third  order  we  have  simply 

W*  _L   iff*      — .    *»»    \ 

'2  "\'3  '!/ 

T       . 

\    A  • 

4/*  (7*       1     o*    )4 

Hence  the  ratios  of  the  triangles  to  the  required  order  become 
I>i?y]      T3 


_    (,a_Ta\_y*V'»       '^T 

„  \s  vi      TS  /       7I~TTTS    TI 


[ 
r, 


;Q)r1T3Vr2    V   ...(42) 


3  (rT 


r./~T12)+    (^  +  ^4    T^T«/' 


where,  if  t1}  t%,  t3  are  the  times  corresponding  to  the  distances  ru  r2,  r3, 

Equivalent  but  rather  simpler  expressions  in  terms  of  the  extreme  distances 
may  be  obtained  by  observing  that 

1_Ji     ^          Ji--JL_?^ 

whence 

rJL_r1       T,       §n»T-BI..l 

3  3  .3'  4^  ^3  3* 

r3       7^1       T3         r2  7'j       7^3 

By  substitution  in  (41)  it  is  easily  found  that 


(43) 


'_    T   3 


From  the  method  by  which  all  the  expressions  of  this  kind  have  been  derived 
it  is  clear  that  the  results  apply  equally  to  all  undisturbed  orbits,  elliptic  or 
hyperbolic. 


CHAPTER   VI 

THE    ORBIT    IN    SPACE 

63.  Hitherto  we  have  considered  the  relative  motion  of  two  bodies  only 
as  referred  to  axes  in  the  plane  in  which  the  motion  takes  place.  It  is  now 
necessary  to  specify  the  manner  .  in  which  the  motion  in  space  is  usually 
expressed. 

We  take  a  sphere  of  arbitrary  unit  radius  with  the  Sun  at  its  centre. 
The  ecliptic  for  a  given  date  is  a  great  circle  on  this  sphere.  That  hemi- 
sphere which  contains  the  North  Pole  of  the  Equator  may  be  called  the 
northern  hemisphere.  On  the  ecliptic  is  a  fixed  point  7  which  represents 
the  equinoctial  point  for  the  given  date  and  from  which  longitudes  are 
reckoned  in  a  certain  direction.  The  plane  of  the  orbit  is  also  represented 
by  a  great  circle  which  intersects  the  ecliptic  in  two  points.  One  of  these 
fl  corresponds  to  the  passage  of  the  moving  body  from  the  southern  to  the 
northern  hemisphere  and  is  called  the  ascending  node ;  the  other  node  is 
called  the  descending  node.  The  longitude  of  fl,  or  7!!,  may  be  denoted  also 
by  n :  it  is  an  angle  which  may  have  any  value  between  0°  and  360°.  The 
angle  between  the  direction  of  increasing  longitudes  along  the  ecliptic  and 
the  direction  of  increasing  true  anomaly  along  the  orbit  is  called  the  in- 
clination and  may  be  denoted  by  i.  It  is  an  angle  which  may  lie  between 
0°  and  180°. 

Let  P  be  the  point  on  the  great  circle  of  the  orbit  which  represents  the 
radius  vector  through  the  perihelion  and  Q  any  other  point  on  the  same 
great  circle  representing  a  radius  vector  with  the  true  anomaly  w,  so  that 
PQ  =  w.  We  may  denote  the  arc  HP  lying  between  0°  and  360°  by  »,  so 
that  OQ  =  w  +  w.  This  angle,  reckoned  from  the  ascending  node  to  any 
point  on  the  plane  of  the  orbit,  is  called  the  argument  of  the  latitude.  It  is 
possible  to  regard  w  as  an  element  of  the  orbit,  but  it  has  been  more  usual 
to  define  the  element  -or,  which  is  called  the  longitude  of  perihelion,  as  the 
sum  of  the  two  angles  fl  +  w  although  only  one  of  these  is  measured  along 
the  ecliptic.  The  angle  CT  +  w  or  D  4-  w  +-  w  is  called  the  longitude  in  the 
orbit.  We  have  thus  defined  the  three  elements,  the  longitude  of  the 
p.  i>.  A.  5 


66 


The  Orbit  in  Space 


[CH.  VI 


ascending  node,  the  inclination  of  the  orbit  and  the  longitude  of  perihelion, 
required  to  fix  the  position  of  the  orbit  in  space,  and  with  these  it  is 
necessary  to  mention  the  date  of  the  ecliptic  and  equinox  to  which  they 
are  referred. 

64.  The  motion  must  now  be  definitely  related  to  the  time.  Let  t0  be 
an  epoch  arbitrarily  chosen  and  T  the  time  of  perihelion  passage.  Then, 
n  being  the  mean  motion,  the  mean  anomaly  corresponding  to  the  epoch  is 

M0  =  n(t0-T). 

Either  M0  or  T  might  be  regarded  as  an  element  of  the  orbit,  but  in  the 
case  of  a  planetary  orbit  it  is  more  usual  to  employ  the  mean  longitude  at 
the  epoch,  e,  which  is  defined  as  the  sum  CT  +  M0.  Thus  at  any  time  t,  if 
u  =  -GT  +  w  is  the  longitude  in  the  orbit  and  E  the  eccentric  anomaly,  the 
position  of  the  planet  is  given  by 


where 


-esmE  =  M  =  n(t-  T) 


=  n  (t  —  t0)  +  e  -  -or. 
The  mean  motion  and  the  mean  distance  are  connected  by  the  relation  (§  24) 

no?  =  /jf  =  k"  (1  +  m)a 

where  m  is  the  mass  of  the  planet  (negligible  in  the  case  of  minor  planets). 
The  complete  elements  can  now  be  enumerated  and  illustrated  by  the  case  of 

the  planet  Mars  : 

Mars  (m  =  1/3  093  500) 

to         1900  Jan.  0,  Oh  G.M.T. 
e         293C 

«•         334 


Epoch       

Mean  longitude 
Longitude  of  perihelion 
Longitude  of  node 

Inclination         

Eccentricity        

Mean  motion     

Log  of  mean  distance 


44'  51"-36 
13      6  -88 

n  48    47      9  -36 

i  1    51      1  -32 

e  0-093  308  95 

n  1886"-51862 

log  a  0-182897033 


Equinox 
1900.0 


The  number  of  independent  elements  is  six,  corresponding  to  the  six  con- 
stants of  integration  which  enter  into  the  solution  of  the  equations  of  motion, 
these  being  in  their  general  form  three  in  number  and  of  the  second  order. 

When  the  orbit  is  parabolic  the  eccentricity  is  1  and  the  mean  distance 
is  infinite.  The  scale  of  the  orbit  is  indicated  by  the  perihelion  distance  q 
and  the  time  of  perihelion  passage  T  is  given  instead  of  the  mean  longitude 


63-65]  The  Orbit  in  Space  67 

at  a  chosen  epoch.  Thus  preliminary  parabolic  elements  of  Comet  a  1906 
(Brooks)  are  shown  as  follows : 

T  1905  Dec.  22-29263     G.M.T. 

a,  89°  51'  53"-7  ] 

ft  286    24    22  -1  [  1906.0 

i  126    26      7  -3  J 

q  1-296318. 

65.  If  axes  0  (x1}  y1}  z-^)  be  taken  such  that  Ox-^  passes  through  the  node, 
Oy1  lies  in  the  plane  of  the  orbit,  and  Qz±  is  in  the  direction  of  the  N.  pole  of 
the  orbit,  the  coordinates  of  the  planet  (or  comet)  are 

ac1  =  r  cos  (&>  +  w),     y^  —  r  sin  (to  +  w),     z^  =  0 

when  its  true  anomaly  is  w.  Let  the  axes  be  turned  about  Oxl  so  that  Oy^ 
takes  the  position  Oy2  in  the  plane  of  the  ecliptic  and  Oz2  is  directed  towards 
the  N.  pole  of  the  ecliptic.  Then 

#2  =  x\,     2/2 =  y\  cos  i  —  z1  sin  i,     z2  =  zl  cos  i  +  yi  sin  i. 

Next  let  the  axes  be  turned  about  Oz2  so  that  Ox3  passes  through  the  equi- 
noctial point  and  Oy3  is  in  longitude  90°.  Then 

x3  —  x2  cos  ft  —  2/2  sin  ft,     y3  =  y2  cos  ft  +  x2  sin  ft,     z3  =  z2. 
Hence  the  relations  between  (a?3,  ys,  z3)  and  (a?1}  t/j,  ^)  are  given  by 

xl  y1  z^ 

x3          cos  n           —  cos  i  sin  H  sin  t  sin  H 

y3  •       sin  n  cos  i  cos  fi          —  sin  i  cos  fl 

2^3  0  sin  i  cos  i. 

This  scheme  will  give  the  heliocentric  ecliptic  coordinates  of  the  planet. 
It  is  convenient  to  write 

sin  a  sin  A  =  cos  O,     sin  a  cos  A  =  —  cos  i  sin  ft 

sin  6"  sin  B'  =  sin  ft,     sin  b'  cos  .5'=     cos  i  cos  ft 
for  then 

xs  =  r  sin  a  sin  (A  +  w  +  w) 

y3  =  r  sin  V  sin  (5'  +  w  +  w) 
z3=  r  sin  t  sin  (&>  +  w). 

Hence,  if  R,  LI,  Bl  are  the  geocentric  distance,  longitude  and  latitude  (the 
last  always  a  very  small  angle)  of  the  Sun,  which  may  be  taken  from  the 
Nautical  Almanac,  and  A,  \,  /3  are  the  geocentric  distance,  longitude  and 
latitude  of  the  planet, 

A  cos  X  cos  /3  =  R  cos  L^  cos  BI  +  r  sin  a  sin  ( A'  +  «o  +  w) 
A  sin  A,  cos  /3  =  R  sin  £j  cos  J5a  +  r  sin  6  sin  (Bf  +  <a  +  w) 
A  sin  /3  =  jR  sin  Bl  +  r  sin  t  sin  (&>  +  ?#) 

whence  the  geocentric  ecliptic  coordinates  of  the  planet. 

5—2 


68  The  Orbit  in   Space  [OH.  vi 

66.  Were  the  elements  given  with  reference  to  the  equator  instead  of 
the  ecliptic,  and  this  is  sometimes  done  (though  not  often),  the  same 
formulae  would  give  equatorial  coordinates  with  the  substitution  of  R.A.  and 
declination  for  longitude  and  latitude.  To  obtain  equatorial  coordinates 
from  ecliptic  elements  another  transformation  is  necessary.  Let  the  last 
system  of  axes  be  turned  about  Ox3  so  that  Oys  comes  into  the  plane  of  the 
equator  and  the  new  axis  Qz±  is  directed  towards  the  N.  pole  of  the  equator. 
Then  the  obliquity  of  the  ecliptic  being  denoted  by  e0, 


y3  sne0. 

From   the   above  relations   between  (x3,  y3)  z3)   and   (xl,  y1}  z^   it   follows 
that  (#4,  2/4,  zt)  and  (x1}  ylt  Zj)  are  related  by  the  scheme  : 

xl  yl  z, 

x4          sin  a  sin  A          sin  a  cos  A          cos^a 
2/4          sin  b  sin  B          sin  b  cos  B  cos  b 

z4  sin  c  sin  C  sin  c  cos  G  cos  c 

where  it  is  easily  seen  that 

sin  a  sin  A  =      cos  ft 
sin  a  cos  A  =  —  cos  i  sin  ft 
cos  a  =      sin  i  sin  II 

sin  b  sin  B  =      cos  e0  sin  ft 
sin  b  cos  B  =      cos  e0  cos  i  cos  ft  —  sin  e0  sin  i 
cos  b  =  —  cos  e0  sin  i  cos  ft  —  sin  e0  cos  t 

sin  c  sin  G  =      sin  e0  sin  II 
sin  c  cos  C  =      sin  e0  cos  i  cos  ft  +  cos  e0  sin  i 
cos  c  =  —  sin  e0  sin  i  cos  ft  +  cos  e0  cos  i. 

The  heliocentric  equatorial  coordinates  of  the  planet  now  become 
#  4  =  r  sin  a  sin  (  A  +  co  +  w) 
2/4  =  r  sin  6  sin  (B  +  to  +  w) 
zt  =r  sin  c  sin  (C  +  to  +  w). 

Thus,  for  example,  the  above  elements  for  Comet  a  1906  lead  to 
x,  =  r  [9-803389]  sin  (243°  29'  42"'3  +  w) 
2/4  =  r  [9-999830]  sin  (331    33    15'1+w) 
z4  =  r  [9-887772]  sin  (  60    14    19  '5  +  w) 
referred  to  the  equator  of  1906'0. 

Let  (ac,  y,  z)  be  the  geocentric  equatorial  coordinates  of  the  planet  and 
(X,  Y,  Z)  the  corresponding  geocentric  coordinates  of  the  Sun,  which  may 
be  taken  directly  from  the  Nautical  Almanac  or  other  ephemeris.  Thus 


66,  67]  The  Orbit  in  Space  69 

But 

ar  =  A  cos  a  cos  8,     y  =  A  sin  a  cos  S,     z  =  A  sin  & 

where  A,  a,  S  are  the  geocentric  distance,  right  ascension  and  declination  of 
the  planet.    These  coordinates  can  therefore  be  calculated  from  the  equations 

A  cos  a  cos  8  =  X  +  r  sin  a  sin  (  A  -f  a>  +  w) 
A  sin  a  cos  8  =  Y  +  r  sin  b  sin  (B  +  w  +  w) 
A  sin  8  =  Z  +  r  sin  c  sin  ((7  +  a>  +  w). 

This  form  of  equations,   introduced   by    Gauss,  is  very  convenient  for  the 
systematic  calculation  of  positions  in  an  orbit. 

67.  The  direct  transformation  of  the  elements  from  one  plane  of  refer- 
ence to  any  other  may  be  made  as  follows.  Let  yAB  represent  the  first 
plane  of  reference,  ^AC  the  second  plane  and  BCP  the  plane  of  the  orbit. 
The  first  set  of  elements  are  yB  =  ft,  BP  =  w  and  180°  -B  =  i.  The  new 
elements  are  7^=  ft',  CP=w,  and  C=i'.  Also  the  position  of  the  new 
plane  of  reference  relative  to  the  old  may  be  defined  by  yA  =  fl1}  A  =^  and 
the  arbitrary  origin  71  by  ^  A  =  ft0.  Hence  the  sides  and  angles  of  the 
triangle  ABC  are 

a  =  &>  —  &>',      b  =  ft'  —  ft0,      c=fl  —  Hj 

A  =  i1}  5  =  180°-;,     <?  =  ;'. 

Now  the  analogies  of  Delambre  may  be  written  in  the  single  formula,  easily 
remembered, 

sin  {45°  +  (45°  -  £6  Ta)}      sin  [45°  +  (45°  -  £ 
sin{45°±(45°-£c)}  cos  |45°  +  (45°  - 


where  the  ambiguities  +  +  must  be  read  consistently  but  independently  in 
two  sets  of  three.  Hence  taking  (1)  all  lower  signs,  (2)  all  +  signs,  (3)  all 
—  signs  and  (4)  all  upper  signs  in  the  above  formula,  we  have 

sin  i  (ft'  -  ft0  +  &>  -  &)')  sin  \i!  =  sin  J  (ft  -  ft^  sin  £  (i  +  i^ 
cos  |  (fl'  —  fl9  4  w  —  «')  sin  £  i'  =  cos  ^  (H  —  ftj)  sin  ^  (t  —  i,) 
sin  £  (O'  —  fi0  —  <y  +  a)')  cos  \i'  =  sin  £  (H  -  Oj)  cos  £  (i  +  t\) 

cos  ^-  (n'  —  n0  —  w  +  co')  cos  ^iv  =  cos  |(n  —  HJ)  cos  ^  (i  —  tj). 

These  formulae  will  serve  directly  if  for  example  it  is  required  to  refer  the 
elements  of  a  minor  planet  to  the  plane  of  Jupiter's  orbit  instead  of  to  the 
ecliptic.  Or  again,  if  ft,  ay  and  i  are  the  elements  referred  to  the  ecliptic 
and  equinox  at  the  date  T  and  ft',  CD'  and  i'  the  elements  for  the  equinox 
T+t,  we  may  put  ftj  =  Hj,  z\  ='  T^  and  ft0  =  Ha  +  i/^  where  fa  is  the  general 
precession.  Hence  when  these  quantities  are  known  the  effect  of  precession 
is  given  by 

tan  £  (ft'  —  Hj  —  ^  -  Aw)  =  tan  \  (ft  -  Hj)  sin  \  (i  +  Tr^/sin  £  (i  —  ir^ 
tan  ^  (ft'  —  H!  —  T/T!  +  Aw)  =  tan  ^  (ft  —  Hj)  cos  £  (t  +  ir^/cos  \  (i  —  TT^ 


70  The  Orbit  in  Space  [CH.  vi 

where  A&>  =  a>  —  (a,  and  (by  Napier's  analogy  involving  B  +  C  and  A) 

...         cos^(n  +  n/-2n1-^1)^ 

tan  %(i-i'}  =  -         —  —  —  —  -          V-  tan  A  7^. 
cos  i  (Q  —  lr  4-  ^i) 


68.  When  the  interval  t  is  moderately  short,  however,  these  rigorous 
equations  for  the  effect  of  precession  are  not  required  and  it  is  more  con- 
venient to  use  differential  formulae.  We  now  consider  <yAB  as  the  fixed 
ecliptic  of  1850.0  and  y^AG  as  a  variable  ecliptic.  Since 

cos  G  =  sin  A  sin  B  cos  c  —  cos  A  cos  B 
—  smC.dG  =  (cos  A  sin  B  cos  c  +  sin  A  cos  B}  dA  —  sin  A  sin  5  sin  c  .  c?c 

=  sin  C  cos  6  .  c?  A  —  sin  a  sin  B  sin  (7cfo  ' 
or 

dC=  —cosb.dA  +  sin  a  sin  5.  dc  .................................  (1) 

Also,  since 

sin  (7  sin  b  =  sin  5  sin  c 
sin  (7  cos  b  .  db  =  sin  B  cos  c.dc  —  cos  C  sin  6  .  d(7 

=  sin  B  (cos  c  —  cos  G  sin  a  sin  b)  dc  +  cos  (7  sin  b  cos  6  .  dA 
or 

sin  (7  .  db  =  cos  G*  sin  b  .  dA  +  sin  B  cos  a  .  cfc     ........................  (2) 

Similarly,  since 

sin  G  sin  a  =  sin  A  sin  c 

sin  (7  cos  a.da=  cos  J.  sin  c  .  dA  +  sin  J.  cos  c.dc  —  cos  (7  sin  a  .  dC 
=  (cos  A  sin  c  +  cos  G  sin  a  cos  6)  d  A 

+  (sin  ,4  cos  c  —  sin  J.  cos  C  sin  a  sin  6)  dc 

=  cos  a  sin  b  .  dA  +  sin  A  cos  a  cos  6  .  dc 
or 

sin  C  .  da  =  sin  b  .  dA  +  sin  A  cos  6  .  c?c    ...........................  (3) 

By  a  slight  change  of  notation  we  now  put  I10,  co0  and  i0  for  the  elements 
at  T=  1850.0,  £1,  &)  and  *  for  the  elements  at  time  T  +  1  (instead  of  H',  &>' 
and  i')  and  define  the  position  of  the  ecliptic  and  equinox  at  T  +  t  relative  to 
those  at  T  by  flj  =  II,  ^  =  TT  and  O0  =  II  +  ty,  so  that 

a  =  G)O  —  &>,    b  —  n  —  n  —  -»/r,    c  =  n0  —  ii 

4  =  7r,  5=  180°  -to,         (7=*. 

Hence  by  substitution  in  (1),  (2)  and  (3) 

cfo'  =      —  cos  (ft  —  II  —  i/r)  C^TT  —  sin  (&>0  —  «o)  sin  i0  .  dll 
sin  t  .  d  (H  —  II  —  -^r)  =  cos  t  sin  (ft  —  II  —  i/r)  cfor  —  cos  (&>0  —  &>)  sin  z'0  .  dH 

—  sin  i.dw=         sin  (ft  —  II  —  -^r)  dir  —  cos  (H  —  II  —  ->/r)  sin  TT  .  dll. 


67-69]  The  Orbit  in  Space  71 

But  in  the  coefficients  of  dYl  we  may  put  i  =  ia,  w  =  &>„  and  TT  =  0,  this  being 
the  mutual  inclination  of  the  fixed  and  moving  ecliptic.  Hence  we  have 
simply 

di  fdt  =  -  cos  (O  -  II  -  i/r)  dirfdt 

d£l/dt  =  d-^r/dt  +  cot  i  sin  (£1  —  II  —  i/r)  dir/dt 
dw  /dt  =  —  cosec  i  sin  (H  —  II  —  i|r)  dTr/dt. 

These  are  to  be  integrated  between  t  =  ^  and  t  =  t2,  and  the"  coefficients  of 
d^/dt  are  variable  with  the  time.  Provided  the  interval  is  no  more  than  a 
few  years,  it  is  sufficiently  accurate  to  proceed  thus.  Writing 

4   =  *i  —  (t'2  —  O  cos  (O  —  II  —  i/r)  dir/dt 

O2  =  H,  +  (t.2  -  <,)  {d-fr/dt  4-  cot  i  sin  (fl  -  II  -  "</r)  efrr/dfc} 

6>2  =  &>!  —  (f2  —  £1)  cosec  i  sin  (O  —  IT  —  i/r)  dirjdt 

we  take  II  +  i/r,  dTr/dt  and  dty/dt  from  appropriate  tables  (e.g.  Bauschinger's 
Tafeki,  No.  xxx)  with  the  argument  T  +  ^  (4  +  ^).  With  0  =  11!  and  i  =  ^ 
approximate  values  of  H2,  i2  can  be  obtained  and  the  calculation  is  then 
repeated  with  the  corresponding  values  |  (f^  +  fl2),  |  (^  + 1'2)  substituted  for 
O  and  i. 

69.  It  is  impossible  to  correct  the  first  observations  of  a  moving  body 
for  parallax  in  the  ordinary  way  because  its  distance  is  unknown.  But  the 
line  of  observation  intersects  the  plane  of  the  ecliptic  in  a  certain  point, 
called  by  Gauss  the  locus  /ictus,  the  position  of  which  can  be  calculated.  If 
the  observation  is  then  treated  as  though  made  from  this  point  the  effect  of 
parallax  is  allowed  for  and  also  the  latitude  of  the  Sun. 

Let  the  observation  be  made  at  sidereal  time  T  at  a  place  whose  geo- 
centric latitude  is  <jj.  Let  a,  S  be  the  observed  R.A.  and  declination,  reduced 
to  mean  equinox.  The  geocentric  equatorial  coordinates  of  the  place  of 
observation  are  (p  cos  0  cos  T,  p  cos  (f>  sin  T,  p  sin  <£),  p  being  the  Earth's  radius 
at  the  place,  and  the  corresponding  ecliptic  coordinates  (phlt  ph2,  ph3),  where 

Aj  =  cos  I  cos  b  =  cos  <£  cos  T 

h2  =  sin  I  cos  b  =  cos  </>  sin  T  cos  e0  +  sin  <£  sin  e0 

hz  =  sin  b          =  sin  0  cos  e0  —  cos  (f>  sin  T  sin  e0 

e0  being  the  obliquity  of  the  ecliptic  and  I,  b  the  longitude  and  latitude  of 
the  Zenith.  Similarly 

H!  =  cos  X  cos  ft  =  cos  8  cos  a 

H2  =  sin  A,  cos  /3  =  cos  8  sin  a  cos  e0  +  sin  B  sin  e0 

H3  =  sin  ^  =  sin  8  cos  e0  —  cos  B  sin  a  sin  e0 

are  the  direction  cosines  of  the  line  of  observation,  A,,  /3  being  the  geocentric 
longitude  and  latitude  of  the  observed  object.  The  Nautical  Almanac  gives 
R!,  LI  and  B1  the  geocentric  radius  vector,  longitude  and  latitude  of  the  Sun. 


72  The  Orbit  in  Space  [CH.  vi 

Hence  in  heliocentric  ecliptic  coordinates  the  equation  of  the  line  of  obser- 
vation is 

x  +  R!  cos  Zj  cos  5j  —  Aap  _  y  4-  RI  sin  L^  cos  ^  —  A2p 

&i  H2 

z  +  -Ri  sin  I?!  —  A3p  _       . 

~ffT 

where  A  is  the  distance  from  the  place  of  observation  to  the  point  (x,  y,  z) 
positively  in  the  direction  away  from  the  object.  If  then  this  line  intersects 
the  plane  of  the  ecliptic  in  the  point  (the  locus  fictus) 

x  =  —  R  cos  L,    y  =  —  R  sin  L,     z  =  0 
A  =  (h3p  -  R,  sin  Bl)IH3 

—  R  cos  L  =  —  R!  cos  L!  cos  Bl  +  p^  —  (hsp  —  R!  sin  B^  H^jH^ 
-  R  sin  L  =  —  R!  sin  L^  cos  B1  +  ph2  —  (h3p  —  Rl  sin  B^  H2/H3. 
But  these  exact  equations  can  be  simplified,  regard  being  had  to  the  small 
quantities  involved.     For  Bl  <  1"  in  general,  so  that  sin  Bl  =  B1>  cos  B^  =  1. 
Also  we  may  put  p  =  pRl  where  p  is  the  solar  parallax,  8"'80.     Hence  writing 
R  =  R1  +  dRl}  L=Ll-{-  dL1}  we  have 


—  cos  Zj  .  d-Rj  +  Rl  sin  L^  .  dLl  ^pR^  —  (h3p  —  B^  R1H1/HS 

—  sin  Zj  .  dRi  —  R!  cos  L^  .  dLl  =pR1h2  —  (h3p  —  J5J  R^^jH^ 
whence 

—  dR1/Rl  =p  (Aj  cos  Zj  +  h2  sin  L^  —  (h3p  —  BJ  (Hl  cos  L^  +  H2  sin  Zx)/  Hs 

dL^  =p  (Aj  sin  Zj  —  A2  cos  L^  —  (h3p  —  Bj)  (Hl  sin  L^  —  H2  cos  Lj)/  H3 
or  again 

—  dRljRl  =p  cos  b  cos  (J^  —  i)  —  (  p  sin  6  —  B^  cos  (Zx  —  X)  cot  ft 

dLi  =  p  cos  b  sin  (7^  —  1)  —  (  p  sin  b  —  BJ  sin  (L^  -  X)  cot  /3 
A  /  R,  =  (  p  sin  b  -  50/sin  /3. 

Here  both  jp  and  Bl  are  naturally  expressed  in  seconds  of  arc.  Thus  d£j  ,  the 
additive  correction  to  the  Sun's  longitude,  is  appropriately  expressed  in  the 
same  unit.  The  Nautical  Almanac  gives  log^,  to  which  the  additive 
correction  is 

j    ,       -r>      dRl      Iog10e        dR^  r  .  ono. 

d  .  log  R,  =  --  .  .,,  =  --  [4-3234  -  !0]. 


Finally,  had  the  observation  actually  been  made  from  the  locus  fictus  it 
would  have  been  made  later  in  time  by  the  interval  required  for  light  to 
travel  the  distance  A.  But  the  light  equation,  or  the  time  over  the  mean 
distance  from  the  Sun  to  the  Earth,  is  4988-5.  Hence  the  additive  correction 
to  the  time  of  observation  is  (in  seconds) 

A      498-5        A 
--'- 


The  reduction  to  the  locus  fictus  is  a  refinement  rarely  employed  in  practice. 


CHAPTER  VII 

CONDITIONS    FOR    THE    DETERMINATION    OF   AN    ELLIPTIC    ORBIT 

70.  There  are  certain  properties  of  the  apparent  motion  of  a  planet  or 
comet  on  the  celestial  sphere  which  bear  on  the  problem  of  determining  the 
true  orbit  and  which  can  be  considered  with  advantage  apart  from  the  details 
of  numerical  calculation  which  are  necessary  for  a  practical  solution.  They 
are  closely  connected  with  the  direct  method  of  solution  devised  by  Laplace, 
but  they  equally  contain  principles  which  are  fundamental  to  all  methods. 

Let  (x,  y,  z)  be  the  heliocentric  coordinates  of  the  planet,  (X,  Y,  Z)  the 
heliocentric  coordinates  of  the  Earth.  Then 


x  =  — 


m  and  m0  being  the  masses  of  the  planet  and  the  Earth.  Let  (a,  b,  c)  be  the 
corresponding  geocentric  direction  cosines  of  the  planet,  so  that 

x  =  X  +  ap,     y=Y  +  bp,     z  =  Z  +  cp  ..................  (1) 

p  being  the  geocentric  distance  of  the  planet.  The  observed  position  of  the 
planet  is  given  in  right  ascension  and  declination  (a,  8),  and  if  the  equatorial 
system  of  axes  be  chosen, 

a  =  cos  a  cos  8,     b  =  sin  a  cos  S,     c  =  sin  8. 
Since 

x  =  X  +  dp  +  Zap  +  ap 

fjuK/r3  —  /j,0X/R3  +  dp  +  2dp  +  ap  =  0 

or 

X 

and  similarly 


74 


Conditions  for  the  Determination 


[CH.  VII 


These  are  three  equations  in  p,  p  and  p  +  f^p/r3,  the  solution  of  which  can  be 
written  down  at  once  in  the  form 


-P 


2/5 


/*/»"-  /Mo/IP 


a    d    X 
b     b     Y 
c     c     Z 

a     d    X 

b     b     Y 

c     c     Z 

add 
b     b     b 
c     c     c 

(2) 


the  value  of  p  not  being  required. 

71.     The  determinants  in  (2)  can  be  calculated  when  the  first  and 'second 
derivatives  of  the  three  direction  cosines  are  known.     Now 

d  —  —  sin  a  cos  8 .  d  —  cos  a  sin  B .  8 

d  =  —  sin  a  cos  8 .  d  —  cos  a  cos  8 .  d2+  2sin  a  sin  8 .  dS— cos  a  cos  8 .  82-  cos  a  sin  8 . 8 


c  =     cos 8.8  —  sin 8 .  82. 

The  derivatives  d,  'd,  8,  8  are  most  simply  calculated  from  a  series  of  observed 
values  by  Lagrange's  interpolation  formulae.  If  the  number  of  observations 
is  three,  made  at  the  times  ti,  t2,  ts,  we  have  according  to  this  rule, 


whence 


(t-t2}(t-t3) 

\vj  ™" —  tg/  \   1  ^~  ^3/ 

2i-L-t, 


al  T 


, 

"2    r 


2  "T 


2o, 


2a« 


-  t,)          (ts  -  t,)  (ts  -  t,) 

or,  if  we  choose  t  =  t2,  the  time  of  the  middle  observation, 


TjTaTs  .  d  =  - 

T1r2T3.d= 
where 


+  T2  (TJ  -  T3)  .  «2  +  T32  .  «3  = 

—  2r2  .  «2  +  2r3  .  a3  = 


Tj2  («„  -  a,)  +    T,2  («3  -  «2) 


==  '2         "1  • 


These  formulae,  which  apply  equally  to  the  declinations,  mutatis  mutandis, 
are  only  correct  if  the  observations  are  made  at  very  short  intervals  of  time 
and  are  ideally  accurate.  Since  the  accuracy  of  observations  has  practical 
limitations,  moderately  long  intervals  must  be  used  and  a  greater  number 
of  observed  places  is  necessary  for  satisfactory  results.  Our  immediate 
concern,  however,  is  rather  with  general  principles  than  practical  methods 
of  calculation. 


70-73]  of  an  Elliptic  Orbit 

72.     It  is  now  possible  to  calculate  the  quantity  I  given  by 


75 


add 

-&2 

a     a     X 

b     b     b 

b     b     Y 

c     c     c 

c     c     Z 

and  we  then  have  by  (2) 


lp  =  (l  +  m0)/R3  -  (1  +  mVr3 


•(3) 


The  mass  of  the  planet,  m,  must  be  neglected  in  a  first  approximation  to  the 
orbit  and  this  is  one  relation  between  p  and  r.  In  essence  it  is  fundamental 
in  all  general  methods  of  finding  an  approximate  orbit.  A  second  relation 
is  available  because  we  know  the  angle  ^r  between  R  and  p,  namely 


r2  =  Rz  +  p*  +  2Rp  cos  -^ (4) 

while  the  projection  of  R  as  a  vector  in  the  direction  of  p  gives 
R  cos  i/r  =  aX  +  bY  +  cZ,     (0  <  ^  <  180°). 

If  r  be  eliminated  between  (3)  and  (4)  an  equation  of  the  eighth  degree  in 
p  results,  and  it  will  be  necessary  to  examine  the  nature  of  the  possible  roots. 
For  the  moment  we  suppose  that  the  appropriate  value  of  p  has  been  found. 
Then  the  corresponding  value  of  p  is  given  by  (2)  and  the  components  of  the 
velocity  can  be  calculated,  since  by  (1) 


x  =  X  +  dp  +  ap,    y=Y+bp 


(5) 


where  X,  Y,  Z  must  be  found  from  the  solar  ephemeris  by  mechanical 
differentiation.  Thus  when  p  and  p  are  known,  (1)  and  (5)  give  the  three 
heliocentric  coordinates  of  the  planet  and  the  three  corresponding  components 
of  velocity  at  a  given  time  t.  From  these  data  the  elements  of  the  planet's 
orbit,  assumed  for  the  present  purpose  to  be  elliptic,  can  be  calculated  without 
difficulty. 

73.  Since  equatorial  coordinates  have  been  used  hitherto,  the  elliptic 
elements  of  the  orbit  will  also  be  referred  to  the  equatorial  plane.  If  new 
coordinates  (£,  tj,  £)  be  taken  so  that  the  axis  of  £  passes  through  the  node 
and  the  axis  of  £  through  the  N.  pole  of  the  orbit,  the  transformation  scheme 
is  (cf.  §  65)  : 

x  y  z 


cos  ft' 

sin  ft' 

0 

-  sin  ft'  cos  i' 

cos  ft'  cos  i' 

sin  i' 

sin  ft'  sin  i' 

—  cos  ft'  sin  i' 

cos  i' 

76  Conditions  for  the  Determination  [OH.  vn 

Hence  in  the  plane  of  the  orbit, 

£  =  x  sin  H'  sin  i'  —  y  cos  ft'  sin  i'  +  z  cos  i'  =  0 
£  =  x  sin  fl'  sin  t'  —  y  cos  H'  sin  i'  +  £  cos  t'  =  0 

giving  for  the  determination  of  H'  and  i' 

sin  H'  sin  i'     cos  O' sin  i'        cos  i ' 


(6) 


yz  —  yz  xz  —  xz        xy  —  xy 

Also,  if  u  is  the  argument  of  latitude  (or  rather  of  declination), 

|  =  r  cos  u  =  x  cos  H'  +  y  sin  H'  ..............................  (7) 

and 

rj  =  —  x  sin  O'  cos  i'  +  y  cos  H'  cos  i'  +  z  sin  i' 
or 

r  sin  w  =  z  cosec  t'     ...................................................  (8) 

by  the  above  equation  for  £.  Similarly,  if  V  is  the  velocity  and  %  the  angle 
between  F  and  the  radius  vector  produced, 

£  =  F  cos  (  w  +  x)  =  ar  cos  ft  '  +  y  sin  ft  '  ..................  (9) 

i)  =  Fsin(w  +  %)  =  z  cosec  i'    ...........................  (10) 

Thus  F  and  ^,  as  well  as  r  and  w,  are  determined.  Now  if  w  is  the  true 
anomaly  at  the  point,  the  polar  equation  of  the  orbit  gives 

p  =  r  (1  +ecosw)    ........................  (11) 

pcotx  =  re  sin  w  .....  .  ...........................  (12) 

since  tan  %  =  rdw/dr.     But  the  constant  of  areas  is 

h  =  Vr  sin  %  =  V(/^p)  =  k  \/p     .....................  (13) 

giving  p  and  hence  e  and  w.  The  mean  distance  a  can  be  deduced  from  the 
known  values  of  p  and  e,  or  directly  from  the  relation 

F2  =  2/z/r  -  fju/a  ..............................  (14) 

and  the  mean  motion  n  from  the  equation  //.  =  k2  =  n2a3.  Also  the  element  vr' 
is  given  by  «r'  =  H'  +  w  —  w.  Finally  the  epoch  of  perihelion  passage  is  deter- 
mined by  the  two  equations 


n(t-T)  =  £-esinE  ..............................  (15) 

E  being  the  eccentric  anomaly  at  the  point  of  the  orbit  observed. 

74.  We  now  return  to  the  consideration  of  the  solution  of  equations  (3) 
and  (4),  following  the  method  of  Charlier,  which  gives  the  clearest  view  of 
the  geometrical  conditions  of  the  problem.  The  first  of  these  equations  is 
based  on  the  assumption  that  the  point  of  observation  is  moving  under 
gravity  about  the  Sun.  The  point  which  so  moves  is  in  reality  the  centre 


73-75]  of  an  Elliptic  Orbit  77 

of  gravity  of  the  Earth-Moon  system  and,  strictly  speaking,  the  observations 
should  be  reduced  to  this  point  and  not  the  centre  of  the  Earth.  But  this  is 
a  matter  of  detail  which  our  immediate  purpose  does  not  require  us  to  stop 
and  consider.  Similarly  we  may  neglect  the  mass  of  the  Earth  as  well  as  that 
of  the  planet  and  put  R  —  1  .  Then  the  equations  become  simply 

lp  =  \-\lr*  .......................................  (16) 

r2  =  1  +  2/>  cos  ^  +  p2   ...........................  (17) 

where  I  and  ty  are  known.  The  position  of  the  planet  becomes  known  when 
either  p  or  r  has  been  found,  and  it  is  simpler  to  eliminate  p.  Thus 

pj*  =  prs  +  2lr3  (r3  -  1)  cos  ^  +  (r3  -  I)2 
or 

l*r*_(l<>  +  2Zcos-f  +  l)r6+2(£  cos  ^  +  1)^-1  =  0    ......  (18) 

Now  the  coefficient  of  r3  is 

2  (I  cos  ^  +  1)  =  {(1  -  1/r3)  (r2  -  1  -  p2)  + 


which  is  obviously  positive,  whether  r  is  greater  or  less  than  1.  And  the 
coefficient  of  r8  is  essentially  negative.  Hence,  by  Descartes'  rule  of  signs, 
there  are  at  most  three  positive  roots  and  one  negative  root.  The  latter 
certainly  exists  because  the  last  term  is  negative  (the  equation  being  of 
even  degree),  and  two  positive  roots  must  satisfy  the  equation,  namely  +1 
(corresponding  to  the  Earth's  orbit)  and  the  root  required.  There  must 
be  a  fourth  real  root,  and  therefore  in  all  three  real  and  positive  roots,  one 
real  and  negative  root  and  four  imaginary  roots.  But  the  third  positive 
root  may  or  may  not  satisfy  the  problem. 

Now  by  (16)  r  is  greater  or  less  than  1  according  as  I  is  positive  or 
negative.  If  then  the  two  roots  which  are  in  question  lie  on  opposite  sides 
of  1,  the  spurious  root  can  be  detected  and  a  unique  solution  of  the  problem 
can  be  found.  But  if  they  lie  on  the  same  side,  they  cannot  be  discriminated 
between  in  this  way,  and  an  ambiguity  exists.  If  we  divide  (18)  by  (r  —  1), 
we  obtain 

/(r)  =  l*r6  (r  +  1)  -  (2£r3cos  ^  +  r5  -  1)  (r2  +  r  +  1)  =  0. 
Thus 

/(0)  =  +  1,    /(+  1)  =  2^-3  cos  T/T) 

so  that  the  roots  are  separated  by  +1,  and  a  unique  solution  exists,  if 
1(1  —  3  cos  i/r)  is  negative. 

75.  The  geometrical  interpretation  is  instructive.  The  equation  (16) 
for  different  values  of  the  parameter  I  represents  a  family  of  curves  in  bipolar 
coordinates,  the  poles  being  E  (the  Earth)  for  p  and  8  (the  Sun)  for  r.  The 
planet  lies  at  the,  intersection  of  one  of  these  curves  with  a  straight  line 


78 


Conditions  for  the  Determination  [CH.  vn 


75]  of  an  Elliptic  Orbit  79 

drawn  through  E  in  a  given  direction.     But  there  may  be  two  intersections, 
and  this  will  happen  if  /(+  1)  or 

pH  (1-3  cos  i/r)  =  (1  -  1/r3)  {1  -  1/r3  +  f  (1  +/>2-r!)} 

is  positive.     This  expression  changes  sign  when  we  cross  the  circle  r  =  1  and 
again  when  we  cross  the  curve 

l-l/r3  +  f  (l+p2-r2)  =  0. 

Putting  p2  =  1  +  r2  —  2r  cos  <£  we  get  for  the  polar  equation  of  this  curve  with 
the  origin  at  $ 

4  -  3r  cos  0  =  1/r3    ...........................  (19) 

or  in  rectangular  coordinates, 


showing  that  the  curve  has  an  asymptote  3#  =  4.  Moving  the  origin  to  E 
we  find  at  once  that  E  is  a  node,  the  tangents  being  y=  ±  2x.  The  whole 
curve  consists  of  a  loop  crossing  the  SE  axis  at  the  point  r  =  '5604,  <£  =  TT,  and 
an  asymptotic  branch,  and  is  shown  as  the  "  limiting  "  curve  in  the  figure. 
The  plane  of  the  figure  is  that  containing  S,  E  and  P  (the  planet);  it  is 
only  necessary  to  show  the  curves  on  one  side  of  the  axis  because  this  is  one 
of  symmetry. 

A  few  curves  of  the  family  (16)  are  also  shown  in  the  figure,  for  values 
of  I  which  indicate  sufficiently  the  different  forms.  When  I  =  0  we  have  the 
circle  r  =  l,  called  here  the  "zero"  circle.  It  is  evident  that  when  I  is 
negative  r  <  1  and  the  curve  lies  entirely  within  the  zero  circle,  while  when  I 
is  positive  r  >  1  and  the  curve  lies  entirely  outside  this  circle.  When  I  has 
a  large  negative  value,  the  curve  consists  of  a  simple  loop  surrounding  S  and 
an  isolated  conjugate  point  at  E.  As  —  I  decreases  from  oo  the  loop  increases 
in  size  until,  when  l  =  —  3,  the  loop  extends  to  E,  where  there  is  a  cusp. 
Afterwards  as  I  approaches  0  the  loop,  still  passing  through  E,  approximates 
more  and  more  closely  to  the  zero  circle. 

When  I  is  positive  the  form  of  the  curves  is  rather  more  complicated.  It 
must  be  remarked  that  I  cannot  be  greater  than  +  3.  For 

I  =  (r3  -  l)/r»p  =  (r~l  +  r~2  +  r~s)  (r  -  I)/  p. 

But  r  >  1  and  r  —  1  <  p.  Hence  the  limit  is  established  and  we  have  only  to 
follow  the  values  of  I  from  +  3  to  0.  At  first  the  curve  consists  of  a  small 
loop  passing  through  E.  As  the  value  of  I  falls  the  loop  expands,  tending 
to  enfold  the  zero  circle.  Finally,  when  I  =  +  0'2959,  it  reaches  the  axis  again 
and  forms  a  node  on  the  further  side  of  S.  As  the  value  of  I  falls  still  further 
the  curve  breaks  up  into  two  distinct  loops.  The  larger  continues  to  expand 
outwards  at  all  points  and  recedes  to  infinity,  while  the  inner,  always  passing 
through  E,  contracts  until  finally  it  becomes  the  zero  circle.  These  features 
in  the  development  of  the  family  of  curves  will  be  evident  in  the  figure. 


80  Conditions  for  the  Determination  [CH.  vn 

It  will  now  be  apparent  that  the  limiting  curve  and  the  zero  circle  divide 
space  into  certain  regions  and  that  the  solution  of  the  problem  of  determining 
an  orbit  by  the  method  indicated  is  unique  or  not  according  to  the  region  in 
which  the  planet  happens  to  be.  Thus  we  distinguish  four  cases  : 

(1)  If  the  planet  is  within  the  loop  of  the  limiting  curve  there  are  two 
solutions. 

(2)  In  the  space  between  the  loop  and  the  zero  circle  the  solution  is 
unique. 

(3)  Outside  the  zero  circle  and  to  the  left  of  the  asymptotic  branch  of 
the  limiting  curve  there  are  again  two  solutions. 

(4)  If  the  planet  lies  to  the  right  of  the  asymptotic  branch  of  the 
limiting  curve  only  one  solution  is  possible.      It  happens  that  newly  dis- 
covered minor  planets  are  usually  observed  near  opposition  and  therefore 
this  is  the  case  which  most  commonly  occurs. 

76.  There  is  another  curve  which  has  considerable  importance  in  the 
problem  of  determining  an  orbit  by  a  method  of  approximation  and  to  which 
Charlier  has  given  the  name  of  the  "  singular  "  curve.  We  may  find  it  thus. 
If  we  eliminate  r  between  the  equations  (16)  and'(17)  we  have 

lp  =  1  -  (1  +  '2p  cos  i/r  +  p2)  ~  ' 

which  is  an  equation  giving  the  values  of  p  for  a  line  drawn  through  E  in 
the  direction  i|r.  Two  of  the  values  become  equal  and  the  line  touches  the 
curve  (16)  if 

I  =  3  (cos  •$>  +  p)(l  +  2p  cos  i/r  +  pn-)~% 

=  3  (cos  i/r  +  p)/r\ 

Hence  the  locus  of  the  points  of  contact  of  the  tangents  from  E  to  the  family 
of  curves  (16)  is 

(l-l/r3)/>  =  3(cos^  +  p)//-3 

or 

2r2(r3-l)  =  3(p2  +  r2-l) 

or  again 

3p2=2r5-  5r2  +  3    ........................  (20) 


This  is   the    equation  of  the  singular  curve.      If  we  change  from  bipolar 
coordinates  to  the  polar  equation  with  the  origin  at  8,  we  obtain 

3  (1  -  2r  cos  <f>  +  r2)  =  Zr5  -  5r2  +  3 

or 

r3  =  4  -  3  cos  <j>/r  ..............................  (21) 

Comparison  of  this  form  with  the  equation  (19)  of  the  limiting  curve  shows 
at  once  that  these  two  curves  are  the  inverse  of  one  another  with  respect  to 


75-77]  of  an  Elliptic  Orbit  81 

the  zero  circle.     From  this  relation  the  form  of  the  singular  curve,  which  is 
shown  in  figure  3,  becomes  apparent. 

The  importance  of  the  singular  curve  arises  thus.  In  general  a  line 
through  E  meets  a  curve  of  the  family  (16)  either  in  one  point  (besides  E) 
or  in  two  distinct  ppints.  In  the  latter  case  the  coordinates  of  the  planet 
are  regular  functions  of  the  time  and  can  be  expanded  in  powers  of  the  time, 
but  each  is  expressed  by  two  distinct  series  between  which  it  is  impossible  to 
discriminate.  When,  however,  the  planet  is  situated  at  a  point  on  the  singular 
curve,  the  two  distinct  series  coalesce  and  each  point  of  the  singular  curve 
corresponds  to  a  branch  point  where  we  may  expect  the  coordinates  of  the 
planet  to  be  no  longer  regular  functions  of  the  time.  This  is  in  fact  the 
case.  Charlier  obtained  the  equation  of  the  singular  curve  by  noticing  that 
along  this  curve  expansion  of  the  coordinates  as  power  series  in  the  time 
ceases  to  be  possible. 

77.  If  the  masses  of  the  Earth  and  of  the  planet  be  neglected,  (2)  may 
be  written  in  the  form 


,         2  3 

where  A^  A.,,  A3  represent  three  determinants  and  l=,&t/k?Al.  It  is  clear, 
as  we  have  already  noticed,  that  r<R  if  I  is  negative  and  r>R  if  I  is 
positive.  Now  the  equation  of  the  plane  of  the  great  circle  tangent  to  the 
;ipparent  orbit  at  (a,  b,  c)  is 


a     a     x 
b     b 


=  0 (23) 


The  coordinates  of  the  Sun  on  the  celestial  sphere  are  (—  X/R,  —  Y/R,  —  Z/R) 
and  of  a  neighbouring  point  to  (a,  b,  c)  on  the  apparent  orbit  (a  +  at  +  ^dtf, 
b+  ...,  c  +  ...).  Hence  the  ratio  of  the  perpendiculars  from  these  points  to 
the  above  plane  is  -  A,/JE  •*•  |£2A:S  =  -  2/lk2t*R.  Thus  I  is  negative  if  the 
Sun  and  the  arc  of  the  planet's  orbit  lie  on  the  same  side  of  the  great  circle 
touching  the  orbit,  and  positive  if  the  Sun  and  the  arc  are  on  opposite  sides. 
In  the  first  case  r  <  R,  in  the  second  r>R.  Hence  we  have  the  theorem 
due  to  Lambert,  which  may  be  expressed  by  saying  that  an  arc  of  the  orbit 
of  an  inferior  planet  appears  concave  to  the  corresponding  position  of  the 
Sun,  but  the  arc  described  by  a  superior  planet  appears  convex.  This  test 
makes  it  immediately  apparent  whether  a  planet  or  the  Earth  is  the  nearer 
to  the  Sun. 

It    may  happen   that   A3    vanishes.     It    is   then    necessary    to    express 
the  coordinates  of  neighbouring   points   on   the   orbit  to  the  third   order 

p.  D.  A.  6 


82  Conditions  for  the  Determination  [CH.  vn 


(a  ±  dt  +  ^tit2  ±  ^i'its,  b  ±  ...,  c  ±  ...).     The  result  of  substituting  in  the  left- 
hand  side  of  (23)  is 


± 


a     a     a 


b     b     b 


and  the  double  sign  shows  that  the  curve  crosses  the  tangent  great  circle.  In 
the  language  of  plane  geometry  there  is  a  point  of  inflexion  on  the  apparent 
orbit.  Now  if  A3  vanishes  either  r  =  R  or  Aj  =  0.  Thus  such  a  point  of 
inflexion  occurs  either  when  a  comet  reaches  the  same  distance  from  the  Sun 
as  the  Earth  or  when  the  great  circle  which  touches  the  orbit  of  a  planet 
passes  through  the  position  of  the  Sun. 

78.  When  the  apparent  orbit  of  a  planet  reaches  a  stationary  point  the 
curve  either  crosses  itself  and  forms  a  loop,  or  without  crossing  itself  it  pursues 
a  twisted  path,  passing  through  a  point  of  inflexion.  At  such  a  point,  as  we 
have  just  seen,  the  tangent  in  general  passes  through  the  Sun.  There  is  a 
related  theorem,  due  to  Klinkerfues,  which  applies  to  the  case  of  a  loop. 
Let  P1}  P2>  P3  be  three  positions  of  the  planet  in  space,  El,  E.2>  E3  the  corre- 
sponding positions  of  the  Earth  and  S  the  position  of  the  Sun.  If  the  first 
and  third  positions  correspond  to  the  double  point  on  the  loop,  E1Pl  and  EXP3 
are  parallel  and  lie  in  one  plane.  Let  SP2  meet  the  chord  PjP3in  p2  and  SE2 
meet  the  chord  E1E3  in  e2.  If  ^  is  the  time  taken  to  describe  PjP2  or  E1E2 
and  t2  the  time  along  P2P3  or  E2E3,t1  :  t2  is  the  ratio  of  the  sectors  SP1PZ, 
SP.2P3  or  very  nearly  the  ratio  of  the  triangles  SP^,  Sp2P3,  that  is 
P\P?.  '•  pzPs-  But  similarly  ^  :  t2  is  nearly  equal  to  the  ratio  E^  :  e2E3. 
Hence  PxP3  and  E1E3  are  divided  by  p2  and  e2  in  approximately  the  same 
ratio  and  therefore  e2p2  is  parallel  to  E1P1  and  ESP3.  Consequently  the 
three  planes  E1SP1,  J£2e2Sp2P2,  E3SP3  have  a  common  line  of  intersection, 
namely  the  line  through  8  parallel  to  E1P1  and  E3P3.  But  on  the  geocentric 
sphere  these  three  planes  correspond  to  three  intersecting  great  circles.  The 
first  and  third  intersect  in  P,  the  double  point  on  the  apparent  orbit.  Hence 
the  great  circle  joining  any  intermediate  point  on  the  loop  to  the  corre- 
sponding position  of  the  Sun  also  passes  through  the  double  point,  at  least 
very  approximately. 

It  may  be  inferred  then  that  if  any  three  points  on  such  a  loop  be  joined 
to  the  corresponding  positions  of  the  Sun,  the  three  great  circles  will  meet  in 
one  point  which  is  also  a  point  on  the  apparent  orbit. 

79.  There  is  some  interest  in  finding  the  geometrical  meaning  of  the 
three  determinants  A1}  A2,  A3  in  (2)  or  (22).  Bruns  has  noticed  that 
A3  =  V3k,  where  k  is  the  geodetic  curvature  of  the  apparent  orbit  on  the 
sphere  and  V  the  velocity  in  this  orbit  at  the  point  (a,  b,  c),  so  that 

V2  =  a?  +  b2  +  c2. 


77-so]  of  an  Elliptic,  Orbit  83 

But  we  shall  now  express  these  determinants  in  terms  of  the  small  circle 
of  closest  contact  or  circle  of  curvature.  This  passes  through  the  points 
(a,  b,  c),  (a  +  dt,  b  +  bt,c+ct)  and  (a  +  at'  +^dt'*,  b  +  ...,  c  +...),  and  the 
equation  of  its  plane  is 

x     y     z     1     =0 

a     b     c     1 
a     b     c     0 

d     b     c     0 
or 

x  (be  -be)  -\-  y  (cd  -  cd)  +  z  (db  —  db)  =  A3  ...............  (24) 

Now 

a?  +  b-  +  c2  =1 

ad  +  bb  +  cc  =  0 

ad  +  bb  +  cc  =  —  V- 

by  successive  differentiation.  Solving  these  as  linear  equations  in  a,  b,  c,  we 
obtain 

aA3  =  be  —  be  —  V"  (be  —  be) 

arid  two  similar  equations.  But  (a/F,  b/V,  c/F)  are  the  direction  cosines  of 
the  point  PI  on  the  tangent  90°  from  (a,  b,  c),  and  the  pole  of  the  tangent  is 
(a0,  60,  c0)  where 

Fa0  =  be  -  be,     F60  =  ca  —  ca,     Fc0  =  ab  -  db 
so  that 

be  —  be  =  aA3  4-  F3a0,  .  .  . 
and 

S  (be  -  6c)a  =  A32  +  F6. 

The  equation  of  the  circle  of  curvature  (24)  becomes  then 

(«  As  +  a0F3)  x  +  (6A3  +  60F3)  y  +  (cA3  +  c0F3)  z  =  A3. 
Hence,  if  &>  is  the  angular  radius  of  this  circle, 

cos2  w  =  A32/(A32  +  F6) 
and  therefore 

A3  =  F3  cot  6. 

This  then  is  the  geometrical  meaning  of  the  third  determinant. 

80.  Next  we  take  A2.  If  (A,  B,  G)  are  the  geocentric  direction  cosines 
oftheSuri,  X  =  -AR,  Y  =  -BR,  Z=  -  CR  and 


A,  =  -  R  [A  (be  -  be)  +  B  (cd  -  ca)  +  C  (ab  -  db)} 
~r 


=  -  R  ~r  {A  (be  -  be)  +  B  (cd  -ca)  +  C  (ab  -  db)} 


"dt 

=  -  RV(Aa0  +  Bb0  +  Cc0)  -  RV (Ad,  +  Bb0  +  Cc0\ 


6—2 


84  Conditions/or  the  Determination  of  an  Elliptic  Orbit  [CH.  vu 

Here  A,  B,  C  are  of  course  constants.  Now  (cr0,  60,  c0)  is  the  pole  P0  of  the 
tangent  at  P,  (a,  b,  c).  The  arc  PP0  passes  through  the  centre  of  the  circle 
of  curvature  and  while  P  is  initially  describing  a  circle  of  angular  radius  o> 
about  this  centre  P0  is  describing  a  circle  of  radius  90°  —  eo  about  the  same 
centre.  If  the  velocity  of  P0,  which  is  in  the  direction  of  the  pole  of  PP0 
opposite  PI,  is  V, 

F'/cos  o>  =  F/  sin  o>,     «„/  V  =  -  «/  F,     &„/  V'  =  -b/  F,     c0/  F'  =  -  c/  F. 

Hence 

A2  =  A,  F/  F  +  P  F  cot  w  (Ad  +  Bb  +  Cc). 
Again 


S  being  the  position  of  the  Sun  on  the  sphere,  and  r  the  perpendicular  arc 
from  S  to  the  tangent  PP,  at  P  to  the  apparent  orbit  (positive  if  drawn  from 
the  same  side  of  PPX  as  P0  or  the  centre  of  curvature).  Also 

Aa  +  Bb  +  Cc  =  Fcos  SP,  =  Fsin  v 

where  v  is  the  perpendicular  arc  from  S  to  the  normal  PP0  to  the  apparent 
orbit  at  P  (positive  if  drawn  from  the  same  side  of  PP0  as  PI).  Hence 

A2  =  —  RVsmr  +  jRF2coto>sini>. 

Thus  the  geometrical  significance  of  the  three  determinants  has  been 
determined  and  we  may  write  (2)  in  the  form 


R  Fsin  r      R  (  V2  cot  o>  sin  v  —  V  sin  r)          F3  cot  to 

which  shows  in  the  clearest  way  how  this  method  of  determining  the  orbit 
depends  on  a  knowledge  of  the  simple  quantities  F,  F,  r,  v  and  a>,  which  can 
be  specified  without  reference  to  any  particular  axes.  To  these  must  be  joined 
the  equation  (4),  which  enjoys  the  same  property. 

It  has  been  remarked  (§  75)  that  I  cannot  be  greater,  than  +  3.     Now 

I  =  AS/^A!  =  -  F2  cot  w/k-R  sin  r. 
Hence  for  a  superior  planet, 

F2  <  3k*R  |  tan  w  sin  r  \ 

which  sets  a  limit  to  the  apparent  velocity  when  w  and  r  are  known,  or  to  the 
curvature  of  the  path  when  F  and  T  are  known. 


CHAPTER  VIII 

DETERMINATION    OF    AN    ORBIT.       METHOD    OF    GAUSS 

81.  Since  a  planetary  orbit  requires  for  its  complete  specification  six 
elements,  it  is  to  be  expected  that  three  positions  of  the  planet,  i.e..  three 
pairs  of  coordinates,  observed  at  known  times,  will  suffice  to  determine  its 
path.  And  this  is  in  general  true,  though  there  are  exceptional  circumstances 
in  which  further  observations  may  be  necessary.  The  formulae  are  a  little 
simpler  when  ecliptic  coordinates  are  employed,  and  though  this  is  not 
essential  we  shall  take  as  the  data  of  the  problem  : 

the  times  of  observation  tlt    £2,    £3 

the  longitudes  of  the  planet  \1}  X2,  X3 

the  latitudes  of  the  planet  &,  $3,  /3s 

the  longitudes  of  the  Earth  Llt  L.2,  L3 

the  Earth's  radii  vectores  R1}  R2,  R3. 

The  angular  coordinates  are  referred  to  a  fixed  equinox  which  will  apply  to 
the  resulting  elements.  The  Earth's  longitude  (which  differs  by  180°  from 
the  Sun's  longitude)  and  radius  vector  can  be  derived  from  the  Nautical 
Almanac  or  other  national  ephemeris  :  the  Earth's  latitude  can  be  neglected, 
or,  if  desired,  allowed  for  by  using  the  method  of  the  locus  fictus  (§  69). 

At  the  time  ti  let  rt  be  the  heliocentric  distance  of  the  planet  and  pf  its 
geocentric  distance.  Referred  to  a  fixed  system  of  rectangular  axes  through 
the  Sun  let  (xi}  y{,  z{)  be  the  coordinates  of  the  planet,  (A{,  Bi}  (7t-)  the 
direction  cosines  of  RI  and  (at,  bi}  Ci)  the  direction  cosines  of  pi}  so  that 


82.     Since  the  three  positions  of  the  planet  lie  in  a  plane  passing  through 
the  Sun 


y\ 

y<i 


or 


=0 


86 


Determination  of  an  Orbit 


[CH.  VIII 


But  (y»z3  —  y3z2)>  (y\z*  —  l/s^i)  and  (y^z.^  —  y^z^  are  the  projections  on  the  yz 
plane  of  the  areas  [r2r3],  [r:jrj]  and  for2].     Hence 

«i  fors]  -  ast  [r,rs]  -f  x,  [r^-,]  =  0 


or 


=  0...(1) 


=  0. . .(2) 
=  0. .  .(3) 


[r2r3](alpl  +  AlR1)-[r1r3] 
And  similarly 

[?-2r<j]  (6j  PJ  +  BiRj}  —  [T^TS]  (b.2p.,  +  B2  R2)  4-  [r^]  (63/o:,  + 
fors]  (c^  +  CiRJ  -  [r!?-3]  (c2p.,  +  C2  R2)  +  [r,ra]  (cs/o3  + 

These  are  the  fundamental  equations  expressing  the  condition  for  a  plane 
orbit.  From  them  one  pair  of  the  six  quantities  pi,  Ri  can  be  eliminated  in 
fifteen  ways.  The  result  immediately  required  is  obtained  by  eliminating 
pl  and  ps,  namely 

where  the  determinants  are  indicated  by  their  first  lines,  from  which  the 
second  and  third  lines  are  to  be  obtained  by  changing  the  letters  without 
changing  the  suffixes,  e.g. 

il     Al 


We  have  now  to  notice  that  these  determinants  are  proportional  to  the 
perpendiculars  to  the  plane 

|    al     x     as      =  0 


y 


C3 


or  the  plane  passing  through  the  points  (alt  bl}  c,),  (a:i,  b3,  c3)  and  the  origin, 
from  the  points  (Al}  Bl}  (7j),  («2,  62,  c2),  (A2,  B2,  C2)  and  (A3,  B3>  C3};  and  these 
are  the  representative  points  of  the  directions  of  Rl}  p2,  R2,  R3  on  the  sphere 
of  unit  radius.  The  perpendiculars  to  the  plane  are  therefore  the  sines  of  the 
perpendicular  arcs  to  the  great  circle  through  (a1}  61}  GJ),  (a3,  bz,  c3)  and  if  these 
arcs  are  B-!,  y82',  B2,  B3  respectively  (due  regard  being  paid  to  sign)  our 
equation  becomes 

for;!]  p2  sin  y32'  =  [r2r3]  Rl  sin  B{  —  fo rs]  R2  sin  B2  +  forj  R3  sin  B3. .  .(4) 


83.  The  points  on  the  sphere  just  named  are  EJ}  E2,  E3,  representing 
the  heliocentric  directions  of  the  Earth  and  lying  on  the  ecliptic,  and  P1}  P2,  P3, 
representing  the  geocentric  directions  of  the  planet.  The  great  circle  men- 
tioned is  PjP3.  Let  this  circle  intersect  the  ecliptic  in  longitude  H2  and  at 
the  inclination  r)2.  Then  we  have  the  same  relation  between  any  one  of  the 
perpendicular  arcs  and  the  longitude  (reckoned  from  H.2)  and  latitude  of  the 
point  from  which  it  is  drawn  as  exists  between  the  latitude  of  a  point  and  its 


82-84]  Method  of  Gauss  87 

right  ascension  and  declination,  the  obliquity  of  the  ecliptic  being  replaced 
by  772 .  That  is  to  say, 

sin  fa'  =  cos  772  sin  fa  —  sin  7/2  cos  /32  sin  (X2  ~  ^2) 
sin  B-i'  =  —  sin  r}.2  sin  (L^  —  H2) 

sin  B2'  =  —  sin  r}2  sin  (L2  —  H2) 

sin  B3  =  —  sin  7/2  sin  (Z3  —  7/2) 

and  as  regards  the  points  P1(  P3 

0  =  cos  7;2  sin  &  -  sin  rjz  cos  fa  sin  (Xj  —  ^2) 
0  =  cos  772  sin  fa  —  sin  7;2  cos  /33  sin  (\3  —  jEQ. 
The  latter  give,  by  addition  and  subtraction, 

2  tan  772  sin  {£  (Xj  +  X2)  —  H.^}  =  sin  (&  -f  /33)/cos  /3j  cos  /33  cos  ^  (X3  —  Xj) 
2  tan  773  cos  {£  (Xj  +  X2)  —  //2}  =  sin  (/3:,  —  /3j)/cos  /Sj  cos  /93  sin  ^-  (X3  —  Xj) 
and  determine  rj.2  and  //.j.     We  now  put 

<?!  =  —  Q  siu  BI/ sin  ft*,     c.2  =  — R2sinB2'/sin  fa',     cs  =  — R3s'm  B3'/sm  fa' 
and 

?i,  =  [>2r3]/[rv3],    ws  =  [n^Mnni]. 

The  equation  (4)  then  takes  the  simple  form 

PZ  =  —  Cj^j  +  c2  —  c3n3. 

Now  this  is  a  purely  geometrical  relation  involving  the  intersections  of  any 
plane  through  the  Sun  with  three  lines  drawn  in  given  directions  through 
the  positions  of  the  Earth.  If  we  imagine  the  plane  to  move  into  coincidence 
with  the  ecliptic,  c1;  c2,  cs  remain  unaltered  while  in  the  limit  plf  p2,  p3  vanish 
and  r1}  r2,  r3  become  coincident  with  Rly  R2,  R3.  Hence  if  we  put 

N,  =  [R.R.^/IR.R,]  =  R2  sin  (La  -  L^jR,  sin  (L, -  LJ 
N3  =  [R,R.2]/[R, R,]  =  R,  sin  (L2  -  LJJR,  sin  (L,  -  L,} 

the  equation 

Q  =  —  clN1+  c2  —  c3N3 

must  be  an  identity,  and  this  can  be  verified.     Hence  by  the  elimination  of  c2 

p2  =  c,  (JV,  -  ?h)  +  c3  (Na  -  O (5) 

which  is  the  required  equation  for  p2. 

84.  Since  fa'  is  the  perpendicular  arc  from  P2  to  P^P3  it  is  geometrically 
evident  that  if  the  observed  arcs  of  the  planet's  orbit  are  of  the  first  order  of 
small  quantities  (and  we  assume  them  to  be  small)  /3./  is  a  quantity  of  the 
second  order.  Hence  the  equation  (4)  shows  that  if  we  are  to  obtain  a  value 
of  p.2  which  is  a  real  approximation  and  not  merely  illusory  we  must  at  the 
outset  employ  values  of  the  ratios  of  the  triangles  which  are  correct  to  the 


88  Determination  of  an  Orbit  [OH.  viu 

second  order  in  the  time  intervals.     Accordingly  we  use  (41)  of  §  61   and 
neglect  the  terms  of  higher  order  than  the  second ;  that  is  to  say, 


where 

It  is  necessary  to  neglect  the  mass  of  the  planet  and  put  p  =  k~:  this  can 
safely  be  done  in  calculating  a  preliminary  orbit,  for  which  the  perturbations 
are  entirely  neglected.  The  equation  (5)  for  /o,  therefore  becomes 


=  fco-W  ..............................................  ,  .....  (8) 

where  k0,  10  are  completely  determined  quantities.  But  if  S2  is  the  angle 
(<  180°)  between  p2  and  Rz  produced, 

r.?  =  R22  +  pJ  +  2R.2p2cosS2  ........................  (9) 

where 

cos  S2  =  cos  P.2  E2  =  cos  /32  cos  (\g  —  L2). 

If  now  />2  be  eliminated  from  (8),  which  corresponds  to  the  definite  form  of 
Lambert's  theorem  (§  77),  and  (9),  an  equation  of  the  eighth  degree  in  r2 
results.  The  nature  of  the  roots  of  this  form  of  equation  has  already  been 
discussed  in  §  74.  But  Gauss  replaced  the  eliminant  by  a  much  simpler 
equation  which  is  easily  found.  We  have 

r2          R2  p.2 


—  -  K  ----  r  —  .          ^  --  - 

sm  o.,     sin  z     sm  (82  —  z) 

where  z  is  the  angle  subtended  by  R2  at  the  planet  in  its  intermediate 
observed  position.     Hence  by  (8) 

RZ  sin  (82  -  z)  '  10  sin3  z 

"       =     " 


or 

/0  sin4  z/R23  sin3  B2  =  -R2  sin  (8S  —  z)  +  k0  sin  z 

and  therefore  if  we  put 

m0  cos  q  =  k0  +  R2  cos  &2 

??in  sin  q  =  R2  sin  £, 

mm0  =  IQJR^  sin3  S2 
where  m0  is  given  the  same  sign  as  10,  we  have  the  simple  form 

m  sin4  z  —  sin  (z  —  q)  ..............................  (11) 


84,  85]  Method  of  Gauss  89 

and  this  is  the  equation  of  Gauss.  This  form  of  equation  does  not  avoid  the 
possibility  of  an  ambiguity  arising  from  two  distinct  roots,  which  is  inherent 
in  the  problem.  But  when  only  one  appropriate  root  exists,  it  is  easily  found 
by  successive  approximation.  In  the  most  common  case,  that  of  a  minor 
planet  observed  near  opposition,  z  —  q  is  small  and  a  first  approximate  value  is 
given  by 

z±  =  q  +  m  sin4  q. 

When  z  is  found  the  corresponding  first  approximations  to  p2  and  n  are  given 
by  (10). 

85.  We  have  now  to  find  the  corresponding  values  of  pl  and  p3.  For 
this  purpose  we  return  to  the  equations  (1),  (2)  and  (3),  and  eliminate 
ps  and  R...  The  result  can  be  written  down  at  once  in  the  form 

or 

where  the  determinants  as  before  are  represented  by  their  first  lines,  the 
other  rows  being  obtained  by  change  of  letters  without  change  of  suffixes. 
Since  the  same  form  of  equation  must  remain  true,  the  directions  of  plf  p2,  ps 
being  preserved,  when  the  plane  of  the  orbit  is  made  to  coincide  with  the 
ecliptic,  in  which  case  pl  =  p.2  —  0  and  n±  becomes  Nlt  the  equation 

N1Rl\Al,a3,A3\  = 
must  be  an  identity.     Hence 

»'iPi  «i,f'3,A3l  =  p,  a2,a3,A 
Now 

'«!,«:>,  A*  '=     cos/SjCosXj     cos/33cosX;>     cosL3 
cos  fii  sin  Xj     cos  /3:!  sin  X3     sin  L3 
sin  &  sin  /33  0 

=  cos  /3X  cos  $3  {—  tan  &  sin  (X3  —  L3)  +  tan  /33  sin  (Xj  —  L3)} 

the  axis  of  z  being  drawn  towards  the  pole  of  the  ecliptic  and  the  axis  of 
x  towards  the  First  Point  of  Aries.  Similarly 

a2,  a3>  As  |  =  cos  /32  cos  /33  {—  tan  /S2  sin  (X3  —  Ls)  +  tan  /3S  sin  (X2  —  L^)} 
and 

|  A1}  as,  A3 1  =  sin  /33  sin  (L^  —  Ls). 
Hence 

where 

,,  _  tan  /32 sin  (X3  —  Z3)  —  tan  #3 sin  (X2  —  Z3) 
tan^  sin  (X3  —  L3)  —  tan  yS3  sin  (X,  —  L3) 

j\f ,  _  RI  tan  /33  sin  (Ls  —  L^) 

tan  /8]  sin  (\.  —  L^)  —  tan  /83  sin  (Xj  —  L3)  ' 


90  Determination  of  an  Orbit  [CH.  vni 

Similarly  the  result  of  eliminating  pl  and  ^  from  the  original  equations  is 
to  give  (interchanging  the  suffixes  1  and  3) 

nsp3  cos  fts  =  M3p2  cosj32  +  (N3-ns)M3    ...............  (13) 

where 

,,  _  tan  #2  sin  (Xj  —  Zj)  —  tan  fa  sin  (X2  —  L\} 
tan  #3  sin  (Xj  —  L^  —  tan  fa  sin  (X3  —  L^) 

,  _  R3  tan  fa  sin  (Lr  -  L3) 


tan  fa  sin  (Xj  —  L^  —  tan  &  sin  (X3  —  LJ  ' 

The  coefficients  Ml}  M^,  M3,  M3  as  well  as  Nlf  N3  are  constants  throughout 
the  process  of  approximation,  but  nl}  n3  must  be  taken  at  this  stage  from  the 
approximate  forms  (6)  and  (7).  Then  (12)  and  (13)  give  values  of  pl  and  p3 
corresponding  to  the  approximate  value  of  p2  already  obtained. 

86.     The  heliocentric  distances,  longitudes  and  latitudes  of  the  planet  are 
next  deduced  by  the  formulae 

Ti  cos  bi  cos  (k  —  L^  =  pi  cos  &  cos  (\i  -  Li)  +  RI 

Ti  cos  bi  sin  (lt  -  Lt)  =  pi  cos  fa  sin  (\  —  Li)'  .........  (14) 

Ti  sin  bi  =  pi  sin  fa 

(i=l,  2,  3),  which  are  at  once  found  by  taking  the  axis  of  x  successively  along 
R1}  Rz  and  R3,  the  axis  of  z  being  always  directed  towards  the  pole  of  the 
ecliptic.  But  these  coordinates  give  the  position  of  the  plane  of  the  orbit,  for 

tan  i  sin  (4  —  H)  =  tan  bl 
tan  i  sin  (ls  —  II)  =  tan  b3 

where  i  is  the  inclination  and  O  the  longitude  of  the  node  ;  or  in  a  form  more 
suitable  for  calculation 

2  tan  i  sin  (^  (^  +  13)  —  li}  =  sin  (6j  +  63)/cos  h  cos  b3  cos  \  (13  —  h) 
2  tan  i  cos  •[£  (^  +  /»)  —  flj  =  sin  (63  —  6j)/cos  6j  cos  63  sin  ^  (/3  —  ^) 

And  now  the  three  arguments  of  latitude  Uj,  giving  the  differences  of  the  true 
anomalies,  can  be  calculated,  for 

tan  Uj  =  tan  (lj  —  fl)  sec  i   ........................  (16) 

(j  =  1,  2,  3).  In  the  case  of  a  comet,  it  is  the  practice  to  take  Uj  <  or  >  180° 
according  as  the  latitude  is  positive  or  negative  ;  in  the  case  of  a  planet,  Uj  is 
placed  in  the  same  quadrant  as  lj—  O.  If  we  calculate  n1}  ns  from 

_  r2  sin  (us  —  w2)  _  r2sin(w2  —  w,) 

rjsin^Mg—  ?/,)'  r3sin(w3  —  M:) 

we  shall  not  obtain  improved  values  of  these  ratios,  because  these  equations 
have  a  purely  geometrical  basis  and  merely  serve  as  a  useful  control  on  the 
accuracy  of  the  calculation  ;  the  values  already  obtained  should  be  reproduced. 


85-8s]  Method  of  Gauss  91 

87.  We  have  now  arrived  at  preliminary  approximations  to  the  values  of 
the  geocentric  distances  pl,  p2,  p3,  the  heliocentric  distances  rlf  r2,  rs  and  the 
arguments  of  latitude  u-^,  u2,  u3.     From  these  quantities  we  might  proceed  to 
deduce  a  complete  set  of  elements.     But  our  results  are  not  accurate  for  two 
reasons  :  (1)  the  effect  of  aberration  has  been  ignored,  and  (2)  the  expressions 
(6)  and  (7)  employed  for  ^  and  n3  were  of  necessity  only  approximate.     The 
effect  of  aberration  may  be  stated  thus.     The  light  observed  at-time  t  left  the 
source  whose  distance  is  p  at  the  time  t  —  A£,  where 

Ai  =  4988-5  p/l  day  =  [776116]  p 

in  days,  498s-5  being  the  light-time  for  unit  astronomical  distance.  Had  the 
source  moved  in  the  interval  A£  uniformly  with  the  velocity  of  the  observer 
at  time  t,  its  position  at  time  t  would  be  correctly  inferred  from  the  observa- 
tion, without  correction,  since  in  that  case  there  is  no  relative  motion  between 
the  source  and  the  observer.  If  now  we  correct  the  observation  for  stellar 
aberration  according  to  tfie  ordinary  rule  the  observer's  motion  attributed  to 
the  source  is  eliminated  and  we  have  the  direction  of  the  observed  body  at 
time  t  —  A£  from  the  observer's  position  at  time  t.  This  is  the  most  convenient 
procedure  in  the  present  case,  because  it  enables  us  to  retain  the  Earth's 
coordinates  (R,  L)  at  the  times  of  observation  t  throughout  the  calculation 
and  to  make  no  subsequent  change  in  the  planet's  observed  coordinates  (A,,  j3) 
supposing  them  to  be  corrected  for  stellar  aberration  at  the  outset.  This 
avoids  many  changes  which  would  otherwise  be  necessary  in  the  calculation  of 
subsidiary  quantities.  It  only  remains  when  approximate  values  of  p  become 
known  to  correct  the  time  t  by  subtracting  A£  in  so  far  as  these  relate  to 
actual  positions  in  the  orbit.  In  particular,  the  corresponding  corrections 
must  be  applied  to  the.  time  intervals  TI}  r2,  r3. 

* 

88.  A  better  approximation  to  the  values  of  nl9  ns  might  now  be  made  by 
using  the  formulae  of  Gibbs  or  those  of  §  62  and  with  these  values  the  whole 
calculation  might  be  repeated.     But  we  proceed  at  once  to  introduce  the 
accurate  formulae  for  the  ratio  of  the  sector  to  the  triangle,  (25)  and  (26)  of 
§  55  in  the  case  of  an  elliptic  orbit.     The  sectors  are 

i  -y  i  [r«.  rs]  ,     \y*  [n  r3]  ,     %ys  |Y,  r2] 
and  are  proportional  to  TI}  r2,  T,,  (now  corrected  for  aberration).     Hence 


Here 

y22  =  w22/(£2  +  si 

y?  ~  V*  =  m.i  (2g.2  -  sin  2#>)/sin3 

by  the  formulae  quoted,  and  in  the  present  notation 
jr.,,  cos  £  (u3  -  u^},      m,?  =  fe^T|8/ 


92  Determination  of  an  Orbit  [CH.  vin 

The  corresponding  equations  for  ylt  y3  can  be  written  down  by  a  symmetrical 
interchange  of  suffixes.  Various  methods  have  been  devised  for  the  convenient 
solution  of  these  equations,  generally  involving  the  use  of  special  tables. 

In  the  absence  of  such  tables,  and  they  are  not  necessary,  we  may  proceed 
thus.     Writing  the  cubic  equation  in  the  form 

f  _  y*  _  ^  Q  (2g)  =  0,     Q  (2g)  =  3  (2g  -  sin  2$r)/4  sin3  ,7 

where  Q  (2g)  approaches  the  value  1  as  g  approaches  the  value  0,  we  compare 
it  with  the  identity 

(X3  -  X-3)  -  3  (X  -  AT1)  -  (X  -  X-1)3  =  0. 

Thus  y  =  c/(\  -  X-1)  if 

c3       _  c2  ^  4m2  Q 
X3-X-3~3~     3 

that  is,  if  c  =  2m  VQ  =  H*-3  -  ^~3)-  Hence  if  X3  =  cot  £/9,  3raV#  =  cot  /3  and  if 
X  =  cot  ^7,  y  =  m\jQ  tan  <y.  But  from  the  other  equation  in  y  we  have 
sin  \g  =  \ll  tan  &  if  y  =  m  cos  8/^/1. 

Accordingly  we  throw  the  equations  in  y  into  the  following  form  : 

cot  /3  = 


cos  8  =  *J(IQ)  tan  7 


sin  \g  =  *Jl .  tan  8 

Then,  calculating  the  function  Q  with  an  approximate  value  g'  of  g,  the  result 
of  solving  these  equations  in  turn  is  to  lead  to  a  new  and  closer  approximation 
g".  With  this  new  value  the  process  is  repeated  until  no  change  is  found 
between  the  initial  and  final  values.  The  true  value  of  g  has  then  been 
arrived  at,  and  finally  (the  value  of  8  being  taken  fr6m  the  last  repetition) 

y  =  m  cos  8/^1. 

Since  2g  is  the  difference  between  the  eccentric  anomalies,  the  first  approxi- 
mation to  its  value  may  be  taken  to  be  the  difference  between  the  true 
anomalies,  that  is,  between  the  arguments  of  latitude.  When  2g  is  small,  as 
it  usually  is  in  the  practical  problem,  the  direct  calculation  of  the  function 
Q  (2g)  is  inaccurate  (cf.  §  34).  But  if  we  write 

log  Q  (20)  =  \4^-  log  sec  fa  -  -Wof-  log  sec  ^ 

the  error  committed  is  practically  negligible  when  2g  <  90°,  and  the  direct 
calculation  only  presents  a  difficulty  when  2g  is  much  smaller  than  this  limit. 
The  verification  of  this  approximate  formula' may  be  left  as  an  exercise. 

It  is  unnecessary  to  repeat  the  solution  of  (19)  until  the  value  of  g  is 
exactly  reproduced.  This  point  may  be  explained  in  general  terms  as  it  is  of 
wide  application.  Suppose  the  equations  to  be  solved  are  y=p(x}>  x  =  q  (y), 
p  and  q  being  any  functions.  These  correspond  to  two  curves  P  and  Q. 
Starting  with  the  approximate  value  xl  we  find  yi=p(®i)  and  hence  (#,,  y^ 


88,  89]  Method  of  Gauss  93 

the  point  Pl  on  P.  Next  we  find  similarly  (x2,  y^  the  point  Ql  on  Q.  This 
gives  the  new  value  x.2  of  x  and  with  this  we  find  successively  (#2,  y2)  the 
point  P2  on  P  and  (x3,  y2)  the  point  Q.2  on  Q.  But  if  the  successive  values 
#],  #2,  #3  do  not  differ  greatly,  the  chords  PiP2,  QiQ?  lie  close  to  the  curves 
P  and  Q  and  their  intersection  nearly  coincides  with  the  intersection  of  the 
curves.  In  this  way  we  find  for  the  correction  to  the  third  value  x.A 

X       X3  =  \X2       X3)  I  \\X%       Xi)       \X3       -^2/i  • 

In  the  above  case  two  solutions  of  (19)  with  application  of  the  correction  just 
indicated  will  generally  suffice  for  the  accurate  determination  of  g  and  y. 

89.  When  the  values  of  y1}  y2,  y-j  have  been  thus  obtained  we  have  new 
values  of  nl  and  n2  by  (17).  The  next  step  is  to  recalculate  p2  by  (5)  and 
Pi,  ps  by  (12)  and  (13).  Hence  r1;  r2,  r3  and  11}  12>  13  by  (14),  new  values  of 
ft  and  i  by  (15)  and  finally  ult  u2)  u3  by  (16).  This  brings  us  back  once  more 
to  the  equations  (18)  in  y.  If  the  result  of  solving  them  with  the  improved 
values  introduced  is  to  leave  n^  and  n3  practically  unaltered,  our  object  is 
attained.  Otherwise  it  is  necessary  to  repeat  the  above  steps  until  a  satis- 
factory agreement  is  reached. 

When  this  stage  has  been  arrived  at  the  problem  has  been  solved,  and  it 
only  remains  to  calculate  the  other  elements  of  the  orbit,  ft  and  i  having 
been  obtained  in  the  last  approximation.  The  three  equations 

p  =  rj{l+ecoB  (v}  -  to)},  (  j  =  1,  2,  3) 

are  linear  in  p,  e  cos  &>  and  e  sin  to.     The  symmetrical  solution  gives 

p  =  T^TS  2  sin  (us  —  w2)/2  r2r3  sin  (u3  —  u2) 
—  e  cos  &)  =  2  r2r3  (sin  u3  —  sin  w2)/2  r2rs  sin  (u3  —  u2) 
e  sin  &)  =  X  r2r3  (cos  u3  —  cos  u^fSt  r2r3  sin  (u-j  —  u.2) 

whence  e  =  sin  (/>,  &>  =  CT  —  O  and  a  =p  see3  <f>.    This,  however,  is  not  the  simplest 
solution.     The  areal  velocity  h  =  fc\/p  (§  26)  and  hence 

A-r,  Vjo  =  [r,  r3]  7/2  =  y,  r,r,  sin  (M»  -  «j)  ..................  (20) 

Thus,  p  being  known,  we  have 

f)      f)  \ 

-  +-  —  2  =  2e  cos  ^  (u,  +  u3  —  2&))  cos  ^  (u3  —  u^ 

Tl     Ts  V  .........  (21) 

IP      f) 

=  2e  sin  £  (wj  +  u3  -  2<o)  sin  ^  (u3  —  i^) 

ri       rs 

which  also  give  e  and  w.  Finally,  if  the  mass  is  neglected,  the  mean  motion 
is  n  =  k"/a3/:i  and  the  mean  longitude  at  the  epoch  tu  is  (§  64) 

e  =  a>  +  £l  +  Ej-e"'BmEj-n(ti-t0)  ..................  (22) 

where 


taji  i  EJ  =  ^j          tan  |  (Wj  -  «),        (j  =  1  ,  2  or  3). 
The  times  tj  are  here  corrected  for  aberration  (§  87). 


CHAPTER   IX 


DETERMINATION    OF    PARABOLIC    AND    CIRCULAR    ORBITS 

90.  The  method  explained  in  principle  in  the  last  chapter  requires  no 
assumption  as  to  the  eccentricity  of  the  orbit.     Its  practical  convenience  is 
greatest,  however,  when  the  eccentricity  is  comparatively  small.     On  the 
other  hand  the  majority  of  comets  move  in  orbits  almost  strictly  parabolic. 
For   these   it   is   important  to  have  approximate  elements  after  the  first 
observations  have  been  secured,  in  order  that  an  ephemeris  may  be  calculated 
to  guide  observers  as  to  the  position  of  the  object.     For  this  purpose  the 
method  of  Olbers  (published  in  1797),  which  depends  on  the  assumption  of  a 
parabolic  orbit,  has  continued  in  use  to  the  present  time.     Although  only 
five  elements  have  in  this  case  to  be  determined  we  still  use  three  complete 
observations  of  the  comet  giving  the  longitude  and  latitude  (Xj,  /3j)  at  the 
three  times  tj.     We  again  take  (Rj,  Lj)  as  the  corresponding  radius  vector 
and  longitude  of  the  Earth  and  pj  the  geocentric  distance  of  the  comet,  so 
that  as  before 

xi  =  aJPj  +  AJ  Rj>     Vi  =  bjPi  +  BJ  RJ>     ZJ  =  c)Pj  +  GjRj- 

Here  (xj,  y/;,  Zj)  are  the  heliocentric  coordinates  of  the  comet,  (dj,  bj,  GJ)  the 
direction  cosines  of  pj  and  (Aj,  Bj,  Cj)  the  direction  cosines  of  Rj.  In  the 
ecliptic  system  of  axes  adopted, 

a,-  =  cos  \j  cos  /3j,     bj  =  sin  Xj  cos  ySj,     GJ  =  sin  fy. 

We  shall  express  p3  in  terms  of  pl  and  for  this  purpose  it  is  possible  to 
eliminate  p.2  and  R2  from  (1),  (2)  and  (3)  in  §  82.  The  same  result  may, 
however,  be  deduced  from  the  condition  that  the  orbit  is  plane  in  another  way. 

91.  If  8  is  the  Sun,  Elt  E2,  Es  the  three  positions  of  the  Earth,  and 
Cj,  C2,  Cs  the  three  positions  of  the  comet,  S,  Clt  C.2,  C3  are  coplanar.      Hence 

[7*1^2]      tetrahedron  SE2dC2 


[r2r3]     tetrahedron 
0 


2  C.A 

0 
B,R, 


0 
C2R, 


a2p.,+A2R2, 
0 


c.2p2+C2R2, 

0 

G2R.^ 
c2p2+C2R2> 


b3ps+B3Rs, 


90-92]    Determination  of  Parabolic  and  Circular  Orbits       95 

A2  B2  C.2  -T-          A»  B. 

i-ttj,  bipi~\-B1  RI,  c\pi-\-C>iRi 


a2  b2  c.2  a3p3+A3R3,  b3p3+B3R3>  c3p3+C3R3 

or,  representing  determinants  by  single  rows, 
[i\f-2]{p3\a3,A2,a2  +R3  A3,A2,  a.2]}  4-[r2r3]  {/^  }  a1}  A2,a2i  +  R,  ^,^1.,,  a2j}  =  0. 

But  if,  leaving  the  directions  of  pl,  p2,  p3  unaltered,  we  move  the- plane  of  the 
orbit  into  coincidence  with  the  ecliptic,  we  see  that  in  the  limit 

[RiR2]R3\  A3,  A2,  a2\  +  [R2R3]  Rl  \  Alt  A2,  «2j  =  0 
must  be  an  identity.     Hence 

[r2r3]   Ich.  A2>  a2\  _     i    \[R2R3]     [rars]|  j  Alt  A2,  az 


a3,  AS,  a2 


Now 


=  Mpl  4-  m. 

ctl}     A2,     c.j     = 
b1}     B2,     b.2 
c        Co      c    I 


cos  \!  cos  ft,     cos  L2,     cos  X2  cos  ft 

sin  Xj  cos  ft,     sin  1/2,     sin  X2  cos  ft 

sin  ft    '  ,         0     ,          sin  ft 


=  sin  ft  cos  ft  sin  (X.2  —  Z2)  —  sin  ft  cos  ft  sin  (Xj  —  Z2) 
and  the  other  determinants  can  be  written  down  by  simple  substitutions. 


Thus 


M  = 


[r2r3]   sin  /Si  cos  /32  sin  (Xa  —  L2)  —  sin  /82  cos  &  sin  (X,  — 


i?%2] '  sin  ft  cos  ft  sin  (X3  —  L2)  —  sin  ft  cos  ft  sin  (X2  —  L2) 

sin  ft  sin  (i/j  —  L.2) 


•(1) 


and 

wi  =  JR 

([^!  R2]      [rjTa]  j  sin  ft  cos  ft  sin  (X3  —  L2)  -  sin  ft  cos  ft  sin  (X2  —  L2) ' 

In  the  practical  problem  the  time  intervals  are  usually  small  and  it  is 
possible  to  substitute  the  ratio  of  the  sectors  for  the  ratio  of  the  triangles, 
both  for  the  comet  and  the  Earth,  so  that 

I  •  2  *S  I  I  -^'2  -^"3  I  tJ  "~"  ^2 

Thus  m  =  0  and  with  sufficient  accuracy  we  may  write 


(3) 


where  M  has  the  value  given  by  (1)  and  (2),  unless  the  comet  is  near  the 
Sun  and  describes  large  arcs  in  comparatively  short  intervals.  The  effects  of 
parallax  and  aberration  are  entirely  neglected. 

92.     The  next  step  is  to  express  rlt  r3  and  the  chord  c  joining  the 
extremities  of  these  radii  in  terms  of  p1.     We  have 

rf  =  2  (a,^  +  A.R,)-  =  pl2  +  R12  +  ^p.R,  cos  ft  cos^  -  LJ    .........  (4) 

r32  =  2  (Ma3p1  +  A3R3y-  =  J/  V  +  R./  +  2MPIR3  cos  ft  cos  (X,  -£,)..  .(5) 


96     Determination  of  Parabolic  and  Circular  Orbits    [CH.  ix 

and 


=  A2pi2  +  (T2  +  2pj  /t<7  cos  </>  .....................  ..............................  (6) 

where 
/      A2  =  2  (Ma,.—  a^f  =  M*  +  1-  ZM  {sin  ft  sin  ft  +  cos  &  cos  ft  cos  (X3  -  X,)} 

g*  =  2  (-4S  Us  -  -4,-R,)2  =  #32  +  Uj3  -  2U,U3  cos  (Ls  -  Z,) 

hgcos<f>  =  R3  \M^a3As  —  "S.a1As\  —  R1  {M'^asAl  —  ^a^A-^ 
=  J/cos  ft  {U3  cos  (\3  —  L3)  -  Rl  cos  (X3  —  Zj)} 

—  COS  ft  {#3  COS  (Xj  —  jL3)  —  .Kj  COS  (Xj  —  Zj)}. 

If  jE'jC'  is  drawn  equal  and  parallel  to  E3C3  it  is  clear  that  CC3  =  E1E3  =  g, 
GC^hp,,  (?!  (73  =  c  and  (7,00,  =  180°  -</>. 

But  Euler's  equation  gives 

6A;  (t,  -  t,)  =  (r,  +  rs  +  of  -  (r,  +  ra  -  cf 

and  this  must  be  satisfied  by  the  appropriate  value  of  pl  in  (4),  (5)  and  (6). 
This  value  must  be  found  by  a  process  of  approximation  and  for  a  suitable 
starting  point  we  may  consider  c  small  in  comparison  with  r^  +  rg,  r1  =  rs 
and  R,  =  1.  Then 

.     6lc  (t3  -  t,)  =  (n  +  rs)f  .  Sc/O-,  +  r,)  =  3  V2  .  c  Vn 
or 

2&2  («,  -  ^)2A2  =  (Pi2  +  2p,  cos  «/>  .  (///n-  ^2)  (Pi2  +  2pi  c°s  &  cos  (A!  -  A)  +  I}4. 
With  approximate  values  of  the  numbers  which  occur  in  this  equation  it  is 
easy  to  find  by  trial  a  value  of  p1  which  is  correct  at  least  to  one  decimal 
place.  Then  with  this  value  of  pl  it  is  possible  to  calculate  c  in  two  ways  : 
(i)  directly  by  (6),  (ii)  through  r1}  r3  given  by  (4)  and  (5)  and  inserted  in 
Euler's  equation,  which  may  be  written  (§  52)  in  the  form 

3fc  («,  -  O/V2  (ri  +  r*f  =  'sin  e'  c  =  2  V2  (ri  +  '»)  sin  o@  V  cos  I0-  •  -(7> 
or  solved  by  special  tables.  Two  values  of  c  thus  correspond  to  a  hypo- 
thetical value  of  pi,  and  the  latter  must  be  varied  until  the  discrepancy 
between  the  former  is  made  to  disappear.  A  rule  analogous  to  that  given  in 
§  88  leads  quickly  to  the  desired  value  of  pj.  For  if  the  values  p/,  p"  lead 
successively  to  the  differences  AjC,  A2c  in  c,  it  is  easy  to  see  that  the  value 
of  P!  to  be  inferred  is  given  by 

Pi  =  Pi"  +  (pi"  -  pi)  A2c/(AlC  -  Age). 
In  ordinary  cases  the  correct  result  is  quickly  obtained  in  this  way. 

93.  When  pl  and  p3  =  Mp^  have  been  obtained  it  only  remains  to  de- 
termine the  elements  of  the  orbit.  The  formulae  of  §  86  arc  again 
appropriate,  namely 

TJ  cos  bj  cos  (lj  —  LJ)  —  pj  cos  fa  cos  (\j  —  LJ)  +  Rj 

TJ  cos  bj  sin  (lj  —  LJ)  =  pj  cos  fy  sin  (\j  -  LJ) 

TJ  sin  bj  =•  PJ  sin  $,- 


92-94]     Determination  of  Parabolic  and  Circular  Orbits      97 

(j  =  1,  3),  for  the  heliocentric  distances,  longitudes  and  latitude  of  the  comet. 
Here  r1}  rz  should  reproduce  the  values  finally  arrived  at  in  the  course  of 
determining  p^.  Also 

2  tan  i  sin  {i  (^  +  13)  —  HI  =  sin  (6j  -I-  63)/cos  6X  cos  63  cos  ^  (13  —  Zx). .  .(8) 


2  tan  i  cos  {£  (^  +  4)  —  O}  =  sin  (63  —  fcj/cos  &!  cos  63  sin  £  (Z3  -  Zj).  .  .(9) 

(0  <  i  <  90°  if  ^  >  /!,  90°  <  i  <  180°  if  13  <  IJ  give  H  and  i.     The  Arguments 
of  latitude  are  given  by 

tan  Uj  =  tan  (lj  —  fi)  sec  i 

(j—l,  3),  where  in  this  case  0  <  Uj  <  180°  if  6;  >  0.     By  the  equation  of  the 
parabola 

yV/  =  \fi\  cos  ^  (  Uj  —  «»)  =  \/r3  cos  ^  (w3  —CD)     ............  (10) 

whence 

0*3  —  V^i     sin  |  («i  -f  M3  -  2o>)  sin  ^  (M3  —  ut) 


V^s  +  Vri       C°S  i  (z*j  +  W3  —  2ft))  COS  i  (w3  —  UT) 
or 

tan  i  (ttl  +  u3  -  2ft>)  =  ^3  ~  ^  cot  {  (u,  -  1^)  ............  (11) 

yr,  -4-  v?'i 

which  gives  &>  =  w  —  H  and  also  5-,  the  perihelion  distance.     Finally,  T  being 
the  time  of  perihelion  passage,  we  have  (§  29) 

T  =  tj-  q*  {tan  ^  (M,-  -  o>)  +  i  tan3  ^  (w,-  -  co)}  A/2/&  .........  (12) 

(j  i  =  1,  3).     This  completes  the  determination  of  the  five  elements. 

94.  It  is  to  be  noticed  that  while  the  first  and  third  observations  have 
been  completely  used,  the  second  observation  has  only  entered  partially  into 
the  calculation.  In  fact  the  five  elements  have  been  determined  from  six 
given  coordinates  in  a  unique  way  because  X2,  ft.2  have  not  been  used 
independently  but  only  in  the  form  cot  /32  sin  (X2  —  X2)  in  the  equation  (1) 
for  M.  Consequently  it  cannot  be  expected  that  the  elements  will  satisfy 
the  second  place  exactly  and  the  magnitude  of  the  discordance  is  an  im- 
mediate test  of  the  derived  orbit.  The  second  place  is  therefore  calculated 
by  finding  (§  29)  w2  =  u2  —  <w  from  (12)  (j  =  2),  r2  =  q  sec2  |w2,  and  hence  the 
coordinates  of  the  comet  by  means  of 

p2  cos  yS2  cos  (X2  —  n)  =  rz  cos  w2  —  R2  cos  (Lz  —  H) 
p.,  cos  /32  sin  (X2  —  O)  =  rz  sin  u2  cos  i  —  R2  sin  (Lz  —  ft) 
p2  sin  /82  =  ?-2  sin  u2  sin  i. 

If  the  residuals  are  small  the  elements  may  be  considered  satisfactory.  If 
the  residuals  appear  large,  on  the  other  hand,  there  are  several  possible 
reasons  for  the  fact.  There  may  be  an  error  in  the  calculation,  there  may  be 
an  error  in  the  observations,  or  the  assumption  of  a  parabolic  orbit  may  be 
unjustified.  The  evidence  of  further  observations  must  be  the  final  test. 
But  without  additional  material  it  is  possible  to  improve  the  orbit  obtained 
p.  D.  A.  7 


98     Determination  of  Parabolic  and  Circular  Orbits    [CH.  ix 

by  reconsidering  the  quantities  which  were  ignored  in  the  course  of  finding 
the  first  elements.  Parallax  and  aberration  may  be  allowed  for.  In  the 
place  of  (3)  may  now  be  written 

Ps  =  p1(M  +  m/pj 

where  M  and  m  are  given  by  (1)  and  the  following  equation.  At  this  stage 
an  approximate  value  of  pl  is  known  and  fr..^]/^^]  can  be  calculated  with 
greater  accuracy  than  by  means  of  (2),  for  example  by  the  application  of  the 
formulae  of  Gibbs  or  by  direct  calculation  of  the  areas,  since  the  sides  of  the 
triangles  and  the  included  angles  are  now  approximately  known.  Thus  the 
approximate  M  in  (3)  can  now  be  replaced  by  the  improved  value  M  +  m/p! 
and  the  remainder  of  the  work  can  be  repeated  from  this  point.  There  are, 
however,  shorter  practical  methods  of  removing  a  discrepancy  in  the  middle 
place,  which  serve  the  purpose  well  enough  since  a  pro  visional,  orbit  is  in 
general  all  that  is  required. 

95.  The  eccentricities  of  planetary  orbits  are  in  general  small  and  hence 
a  circular  orbit  may  prove  a  useful  approximation  to  the  true  path,  just  as  a 
parabolic  orbit  is  a  useful  preliminary  step  towards  the  orbit  of  a  periodic 
comet.  As  the  eccentricity  vanishes  and  the  position  of  perihelion  ceases  to 
have  a  meaning,  the  number  of  elements  to  be  determined  is  reduced  to  four 
and  two  complete  observations  of  position  only  are  required.  Thus  if  a 
minor  planet  has  been  found  on  two  photographs  of  the  sky  and  no  other 
observations  are  immediately  available,  a  search  ephemeris  based  on  a 
circular  orbit  may  be  a  useful  guide  in  examining  other  plates  which  may 
have  been  taken  at  the  same  or  at  other  observatories. 

To  consider  the  problem  in  a  general  form  let  (Xl}  F1(  Z^,  (X2,  Y2,  Z2) 
be  the  geocentric  coordinates  of  the  Sun  at  the  times  of  observation  ti,  t2 
and  let  (11}  m1}  n^,  (12,  m2,  n2)  be  the  direction  cosines  of  the  observed 
directions  of  the  planet.  The  axes  may  be  any  fixed  system  with  the  Sun 
at  the  origin.  The  planet  is  observed  to  lie  on  the  lines 

(x  +  X^/l,  =  (y  +  FO/JW,  =  (z+  Z^/n,  =  Pl 
(x  +  X2)/12  =  (y+  F2)/m2  =  (z+  Z.2)/n*  =  p2 

p1 ,  p2  being  the  geocentric  distances.     Hence,  if  a  is  the  radius  of  the  orbit, 
a?  =  (1IP1  -  XJ  +  (mlpl  -  F,)2  +  (nlpl  -  Z^ 

=  p?  -  2PI  (I.X,  +  m,  Y,  +  n.Z,)  +  X?  +  1?  +  Z* 
=  p22  -  2p2  (l^X2  +  m2Y2  +  n2Z2)  +  X*  +  F22  +  Z2* 
and,  if  n  is  the  mean  motion  and  t2  —  tl—r, 

a2  cos  JIT  =  (llpl  -  XJ  (I2p2  -  X.,)  +  (mlpl  -  Fj)  (m2p2  -  F2)  4-  (ihp^-Zj  (n2p.2-Z2) 
=  p^p2  cos  6  -  p1  (ljX2  +  -Wj  F2  4-  «,^2)  -  pz  (^Xl  +  m.2  Fj  +  /^Z,) 


94,  95]    Determination  of  Parabolic  and  Circular  Orbits       99 

Avhere  0  is  the  angle  between  the  observed  directions.  Since  6  is  a  small 
angle  the  equation 

cos  6  =  l^  +  n^rriz  +  Wi%2 

is  unsuitable  for  its  determination,  but  the  proper  modification  depends  on 
the  choice  of  coordinates.  Similarly  n  cannot  be  accurately  determined 
from  COSWT. 

If  we  now  put 

Al  =  11X1  +  Wj  Yl  +  M^  ,     Az  =  12X2  +  w2  Y.>  +  n2Z2 

B!  =  l^Xz  +  Wj  Y2  +  w^a,     B2  =  12X^  +  m2  Yl  +  n2Zt 
we  have 

a2  =  Pl*  -  2AlPl 


a~  cos  nr  =  p^p2  cos  6  —  B1pl  —  B2p2  +  X^X2  +  Yl  Y2  +  Z1Z2. 
Hence 

4«2  sin2  $nr  =  p^  +  p2-  —  2pip2  cos  6  —  2  (A1  —  BJ  p^  —  Z  (A2  —  B2)  p., 
+  (X.  -  Xtf  +  (Y2-  F,)2  +  (Z,  -  Ztf 

cos2  \e  {p2  -Pl-$(Ai-Al-Ba+  BJ  sec2  %ey 

+  sin2  1  6  [p2  +  pl-$(A*+A1-  B.2  -  B,}  cosec2  &  O}2 


sec^  0  -  \(A2  +  Al-Ba-  5,)2  cosec2  £0. 

The  equations,  which  must  be  solved  by  trial,  can  therefore  be  reduced  to 
the  form 

sini/r^  MJa,     pt  =  a  cos  ^  +  Al  ~\ 

sin  i/r,  =  M2/a,     p2  =  a  cos  fa  +  A2  I...  (13) 

4a3  sin2  |nr  =  cos2  \Q  (p2  —  pl  —  6j)2  +  sin'J  £0  (p2  +  pl  —  62)2  +  cj 
where  (without  the  transformations  -appropriate  to  the  coordinate  system) 
M?  =  X:-  +  I?  +  Z*  -  A?,    MJ  =  X.?  +  F2S  +  Z*  -  A./ 
b,  =  (A.,-A1-  B2  +  A)/2  cos2  \e 
63  =  (A2  +  Al-B2-  5^/2  sin2  £0 
c  =  (X,  -  XJ-  +  (Y2-  Y,r~  +  (Z2  -  Zrf 

-(A2-B2-A,  +  5,)2/4  cos2  ^  0  -  (^2  -  B2  +  A,  -  B^/4,  sin2  \  0. 

A  trial  value  of  a  gives,  by  (13),  fa,  fa  and  hence  pl}  p2;  these  lead  to  a 
value  of  n  and  the  process  is  continued  until  values  are  obtained  consistent 
with  the  relation  w2a3  =  &2.  In  the  case  of  a  minor  planet  log  a  =  0'4  is 
indicated  as  the  appropriate  initial  value.  With  the  above  formulae  the 
calculation  can  be  performed  directly  in  equatorial  coordinates,  and  little 
will  be  gained  by  introducing  the  ecliptic  system.  When  a  and  n  have  been 

7—2 


TOO    Determination  of  Parabolic  and  Circular  Orbits   [CH.  ix 

found,  p-i,  p2  are  also  known  by  (13)  and  hence  the  heliocentric  coordinates  of 
the  planet 

Vi  =  lipi-Xi,     yi  =  m1p1-Yl,     zl  =  n1p1-Zl 

%2  =  4/t>2  -  X*,     2/2  =  intfz  -  YZ,     zz  =  KZP*  -  %z- 

96.  Gauss  has  given  a  method  for  finding  a  circular  orbit,  based  on 
ecliptic  coordinates.  Let  (JR^,  L^,  (R2,  L2)  be  the  heliocentric  distances  and 
longitudes  of  the  Earth  at  the  times  tlt  t2  and  (\,  /3j),  (X2,  /32)  the  cor- 
responding observed  longitudes  and  latitudes  of  the  planet.  If  in  the  plane 
triangle  SE1Pl  the  angle  at  Pj  is  denoted  by  zl  and  the  exterior  angle 
at  E,  by  Blt  P1SE1  =  B1-z1  and 

a  sin  zl  —  RI  sin  ^    (14) 

Similarly  in  the  triangle  SE2P2,  with  similar  notation, 

asin^2  =  R2  sin  82     (15) 

The  directions  of  the  sides  of  the  two  triangles  are  now  represented  on  a 
sphere  of  unit  radius,  SE1}  8E2  being  represented  by  El,  E2  on  the  ecliptic, 
SPl}  SP2  by  two  points  P1}  P2.  If  Gl}  G2  represent  E^P^,  E2P2,  these 
points  lie  respectively  on  the  great  circles  E-1P1,  E2P2  and  the  arcs  E1G1> 
E2G2  are  Sx  and  82.  Let  the  circles  ElGl,  E2G2  cut  the  ecliptic  at  the 
angles  fyl,  y2.  Then  the  projections  of  the  radius  through  GI  on  the  radius 
through  Ely  the  radius  through  the  point  on  the  ecliptic  90°  in  advance 
of  -fc'j  and  the  radius  through  the  pole  of  the  ecliptic  give 

cos  y3j  cos  (A-i  —  jLj)  =  cos  B1 

cos  &  sin  (Xx  —  Lj)  =  sin  Sl  cos  ^ 

sin  ySj  =  sin  81  sin  ^ 

and  similarly 

cos  /32  cos  (X2  —  L2)  =  cos  8.2 

cos  /82  sin  (X2  —  Z/2)  =  sin  &2  cos  7^ 

sin  /32  =  sin  S2  sin  j2 

whence  S1}  S2  and  7^.  72.  Let  the  circles  ElPl)  E2P2  meet  in  D  at  an  angle  77. 
If  DEl  =  (f>l  and  DE^  =  <^.2,  the  analogies  of  Delambre  applied  to  the  triangle 
DE1E2  in  which  the  side  E^EZ  is  Z2  —  L^  and  the  adjacent  angles  are  71,  TT  —  70, 
give 

^  +  <^>2\)         •      (TT  _  fir      TT  —  72  ± 
- "    8      Jf      S 

^_-j2)}'  cos  • 

[44  2     /J  (4 

or  more  explicitly 

sin  ^77  sin  ^  (^  +  </>2)  =  sin  ^  (Z2  - 1^}  sin  £  (7,  +  7,) 
sin  ^77  cos  %  (</>!  +  <^3)  =  cos  £  (Za  -  A)  sin  \  (y2  -  7, 
cos  £ ?;  sin  |  (0!  -  ^)  =  sin  |  (Xa  -  LI)  cos  |  (7.3  +  7 
cos \T) cos |(^!  -  <£2)  =  cos  |  (L2 - L^ cos ^  (73 - 7 


95-97]     Determination  of  Parabolic  and  Circular  Orbits     101 


whence   <£1;    <£„   and    77.     But   since    the   arc    E1P1  =  B1  —  z1   and 

DPl  =  (f>i  —  8i+Zi  and  DP2  =  <p^  —  S.2+  z2>  while  PlP2  =  n  (t.2  —  ^),  n  being  the 

mean  motion.     Hence 


cosn(t2—  tl)=cos(<f)l  —  8l+zl)cos(<f>2—  82+^2)  +  sin(^)1  —  S1+2'1)sin(^)2—  S2+ 
or  better,  since  n  (£2  —  •  £1)  is  a  small  angle, 

sin2  1  re  (£2  -  £j)  =  cos2  £77  sin2  i  (^  +  .sv,  -  s,)  +  sin2  \T\  sin2  £  (x2  +  z^  +  zj..  .(16) 
where 


The  solution  is  conducted  in  the  usual  way.  Since  Slt  S2  are  known  an 
assumed  value  of  a  gives  zly  z«  by  (14)  and  (15).  Then  ^,,  ^2  and  »?  being 
known,  the  value  of  n  is  deduced  from  (16),  and  the  process  is  continued 
until  values  are  found  which  satisfy  the  relation  W2a3  =  &2.  When  this  has 
been  done,  the  values  of  z1}  z»  have  also  been  found,  and  hence  the  geo- 
centric distances  are  given  by 

pl  sin  z-i  =  .Rj  sin  (Si  -  z^),     p2  sin  zz  =  R2  sin  (S2  —  z2) 

but  these  distances  are  not  actually  required.  Since  the  arc  ElPl  on  the 
sphere  is  8l  —  zl  and  makes  the  angle  ^  with  the  ecliptic,  we  have  the 
heliocentric  longitude  and  latitude  of  P1  (as  in  the  case  of  G^)  given  by 

cos  6j  cos  (^  —  Z,)  =  cos  (B1  —  z^) 

cos  6t  sin  (l-i  —  Zj)  =  sin  (Sj  —  ^)  cos  ^l 

sin  bl  =  sin  (Sj  —  z^)  sin  71 

with  similar  formulae  for  (1%,  b2)  the  heliocentric  longitude  and  latitude  of 
the  planet  in  its  second  position. 

97.     If  (/i,  &]),  (12,  12)  have  been  thus  obtained  the  remaining  elements 
are  easily  found.     For  by  (15)  of  §  86"  the  node  and  inclination  are  given  by 

2  tan  i  sin  (i  (^  +  £2)  —  H}  =  sin  (6j  +  b2)/cos  h  cos  62  cos  ^  (12  —  ^) 
2  tan  i  cos  {£  (/,  +  12)  —  n  j  =  sin  (b.2  —  6j)/cos  6j  cos  62  sin  £  (/2  —  /j) 
and  then  the  arguments  of  latitude  by 

tan  MJ  =  tan  (^  —  II)  sec  i,     tan  u2  =  tan  (Z2  —  H)  sec  i 

with  the  check  u2  —  MI=  n.  (^2  —  ^).  As  the  fourth  element  the  argument  of 
latitude  i.i0  at  a  chosen  epoch  t0  may  be  taken,  and  this  is  simply 

u0  =  MJ  +  w  (<0  -  tj)  =  u2  +  n  (t0  —  t2) 
where  tl}  t2  may  be  antedated  for  planetary  aberration. 

If,  on  the  other  hand,  the  heliocentric  coordinates  (xly  yl}  z^  and  (x2,  y2,  z^) 
have  been  found  as  in  §  95,  and  i'  is  the  inclination  of  the  orbit  to  the 


102    Determination  of  Parabolic  and  Circular  Orbits   [OH.  ix 

plane  z  =  0  and  ft'  is  reckoned  in  this  plane  from  the  axis  of  x  towards  the 
axis  of  y,  the  plane  of  the  orbit  is 

x  sin  ft'  sin  i'  —  y  cos  ft'  sin  i'  +  z  cos  i'  =  0 
and  as  this  is  satisfied  by  the  two  points  on  the  orbit  we  have 

sin  ft'  sin  i'  _  cos  ft'  sin  i'  _       cos  i' 
y\zz  —  y%Zi       x\z<2     (K^ZI      Xi  yz     xzy\ 

The  solution  can  then  be  completed  as  before,  the  arguments-  u  being  now 
reckoned  in  the  plane  of  the  orbit  from  the  node  in  the  plane  z=Q. 

The  meaning  of  the  quantities  6j,  b2  and  c  in  §  95  may  be  seen  thus.  Let  an 
axis  of  z  be  taken  perpendicular  to  p1  and  p»,  and  an  axis  of  x  midway  between 
the  directions  of  pl  and  p2,  so  that  (llt  w,,  HI)  become,  (cos  ^6,  —sin  |#,  0), 
(£2,  w2,  n.2)  become  (cos|#,  sin  ^ft,  0),  and  (Xlt  F,,  Z^,  (X.,,  Y»,  Z.2)  become 
(i/,  F/,  *,'),  W,  F/,  £/).  Then 

6,  =  (X/  -  Z,')  sec  |^ 


If  the  difficulties  of  reducing  this  apparently  simple  problem  to  a  practical 
form  of  calculation  are  carefully  considered,  in  view  of  the  small  quantities 
which  occur,  the  merit  of  the  method  in  §  96  will  be  better  understood.  The 
reader  must  realize  that  the  general  problem  of  determining  orbits  from 
observations  close  together  in  time  is  essentially  a  question  of  arithmetical 
technique,  and  not  of  any  particular  mathematical  difficulty.  This  is  well 
illustrated  in  the  history  of  the  problem,  especially  in  the  eighteenth  century. 

It  is  to  be  remarked  that  the  problem  of  finding  a  circular  orbit  to 
satisfy  the  given  observations  cannot  always  be  solved.  That  a  solution  is 
not  necessarily  to  be  expected  with  arbitrary  data  can  be  readily  seen, 
though  the  equations,  not  being  algebraic,  are  too  complicated  to  make  a 
general  discussion  of  the  conditions  feasible.  It  is  enough  to  say  that  cases 
have  occurred  in  practice  in  which  a  circular  approximation  to  the  orbit  has 
proved  impossible.  The  number  of  minor  planets  already  discovered  is 
approaching  a  thousand,  and  the  most  frequent  eccentricity  is  in  the  neigh- 
bourhood of  012. 


CHAPTER  X 

ORBITS    OF    DOUBLE    STARS 

98.  There  exist  in  the  sky  pairs  of  stars  the  components  of  which  are 
separated  by  no  more  than  a  few  seconds  of  arc,  and  frequently  by  less  than 
one  second.  So  close  are  they  that  they  can  only  be  seen  distinctly  in 
powerful  telescopes,  if  indeed  they  can  be  clearly  resolved  at  all.  Such  pairs 
are  so  numerous  that  probability  forbids  the  idea  that  the  contiguity  of  the 
stars  can  be  explained  by  chance  distribution  in  space.  They  must  be 
physically  connected  systems  for  the  most  part  and  it  is  to  be  expected  that 
the  relative  motion  of  the  stars  will  reveal  the  effect  of  mutual  gravitation. 
That  this  is  actually  true  was  discovered  by  Sir  W.  Herschel. 

The  motion  is  referred  to  the  brighter  component  as  a  fixed,  point.  The 
relative  motion  of  the  fainter  component  takes  place  in  an  ellipse  of  which 
the  principal  star  occupies  the  focus  (§  24),  unless  there  are  other  bodies  in 
the  system,  or  there  proves  to  be  no  physical  connexion  between  the  pair. 
The  apparent  orbit  which  is  observed  js  the  projection  of  the  actual  orbit  on 
the  tangent  plane  to  the  celestial  sphere,  to  which  the  line  of  sight  to  the 
principal  star  is  normal,  and  since  the  point  of  observation  is  very  distant 
compared  with  the  dimensions  of  the  orbit  the  projection  can  be  considered 
orthogonal.  Hence  the  law  of  areas  holds  also  in  the  apparent  orbit,  which 
is  equally  an  ellipse.  But  in  this  orbit  the  brighter  star  does  not  occupy  the 
focus :  its  position  gives  the  means  of  determining  the  relative  situation  of 
the  true  orbit. 

The  observations  give  the  polar  coordinates, •  p,  8,  of  the  companion,  the 
principal  star  being  at  the  origin.  The  distance  p  is  expressed  in  seconds  of 
arc  and  the  linear  scale  remains  unknown  unless  the  parallax  of  the  system 
has  been  determined.  The  position  angle  6  is  reckoned  from  the  North 
direction  through  360°  in  the  order  N.,  E.  or  following,  S.,  W.  or  preceding. 
The  planes  of  the  actual  and  apparent  orbits  intersect  in  a  line  called  the  line 
of  nodes  and  passing  through  the  principal  star.  The  position  angle  of  that 
node  which  lies  between  0°  and  180°  will  be  designated  by  H.  Thus  if  the 
line  of  nodes  is  taken  as  the  axis  of  , 


104  Orbits  of  Double  Stars  [CH.  x 

On  the  other  hand,  in  the  plane  of  the  actual  orbit,  the  longitude  of  periastron 
X  is  the  angle  measured  from  this  node  to  periastron  in  the  direction  of 
orbital  motion.  Hence  in  this  plane,  if  the  line  of  nodes  is  taken  as  the  axis 
of  x, 

x  =  r  cos  (w  +  X),     y  =  r  sin  (w  +  X) 

where  r  is  the  radius  vector  and  w  the  true  anomaly  of  the  companion.  But 
if  i  is  the  inclination  of  the  two  planes  to  one  another,  £  =  x  and  v)  =  y  cos  i, 
so  that  . 

p  cos  (6  —  O)  =  r  cos  (w  +  X) 

p  sin  (9  —  H)  =  r  sin  (w  +  X)  cos  *'. 

Here  the  limits  contemplated  for  i  are  0°  and  180°.  If  0°  <  i  <  90°,  6  and  w 
increase  together  with  the  time  and  the  motion  is  direct.  If  90°  <  i  <  180°, 
B  decreases  with  the  time  and  the  motion  is  retrograde.  This  is  a  departure 
from  the  more  usual  convention  according  to  which  i  is  always  less  than  90°. 
It  is  then  necessary  to  state  whether  the  motion  is  direct  or  retrograde,  and 
in  the  latter  case  to  reverse  the  sign  of  cos  i.  Ordinary  visual  observations 
of  double  stars,  however,  must  leave  the  position  of  the  orbital  plane  in  one 
respect  ambiguous,  since  there  is  nothing  to  indicate  whether  the  node  as 
defined  is  the  approaching  or  receding  node.  The  two  possible  planes  intersect 
in  the  line  of  nodes  and  are  the  images  of  one  another  in  the  tangent  plane 
to  the  celestial  sphere. 

In  addition  to  the  three  elements,  fl,  X,  i,  now  defined,  four  other  elements 
are  required.  These  are  a,  the  mean  distance  in  the  true  orbit,  expressed 
like-p  in  seconds  of  arc;  e,  the  eccentricity  of  the  true  orbit;  T,  the  time  of 
periastron  passage ;  and  P,  the  period  (or  n  =  2?r/P,  the  mean  motion)  ex- 
pressed in  years. 

99.  The  measurement  of  double  stars  is  difficult  and  the  early  measures 
were  very  rough  indeed.  As  the  accuracy  of  the  observations  is  not  high 
refined  methods  of  treatment  are  seldom  justified  and  graphical  processes 
have  been  largely  employed.  The  observed  coordinates  may  be  plotted  on 
paper  and  the  apparent  ellipse  drawn  through  the  points  as  well  as  may  be. 
Let  C  be  the  centre  and  S  the  position  of  the  principal  star.  The  problem 
consists  in  finding  the  orthogonal  projection  by  which  the  actual  orbit  is 
projected  into  this  ellipse  and  the  focus  F  into  the  point  S. 

The  direction  of  the  line  of  nodes  can  be  determined  by  the  principles  of 
projective  geometry.  Conjugate  lines  through  the  focus  F  form  an  orthogonal 
involution.  They  project  into  an  overlapping  involution  of  conjugate  lines 
through  S.  Of  this  involution  one  pair  is  at  right  angles  and  as  in  this  case 
a  right  angle  projects  into  a  right  angle  it  is  clear  that  the  line  of  nodes  is 
parallel  to  one  of  the  pair.  Let  SA,  SA' ;  SB,  SB'  be  two  pairs  of  conjugate 
lines  through  8.  When  the  apparent  ellipse  has  been  drawn  these  can  be 


98,  99] 


Orbits  of  Double  Stars 


105 


found  by  drawing  tangents  at  the  extremities  of  chords  through  S;  or  by 
inscribing  quadrangles  in  the  ellipse,  for  each  of  which  S  is  a  harmonic  point. 
On  CS  as  diameter  describe  a  circle,  centre  K.  Let  Al}  AJ ;  Blt  B^  be  the 
points  in  which  the  conjugate  lines  intersect  th"is  circle  and  let  A1Al',  BlBl' 
intersect  in  0.  Corresponding  points  of  the  same  involution  on  the  circle 
are  obtained  by  drawing  chords  through  0,  and  if  OK  meets  the  circle  in 
N,  N',  SN,  SN'  are  the  orthogonal  pair  of  the  involution  pencil  required. 
Let  CABNA'B'  be  a  transversal  of  the  pencil  drawn  parallel  to  SN'  so  that 
A  A',  BB'  subtend  obtuse  angles  at  8.  This  is  an  involution  range  of  which 
N,  since  it  corresponds  to  the  point  at  infinity,  is  the  centre,  so  that 
AN.  NA'=BN .  NB'.  On  NS  take  the  point  F  such  that  NF*  is  equal  to 
this  constant  product.  Then  F  is  the  intersection  of  circles  on  the  diameters 
A  A',  BB'  and  A  FA',  BFB'  are  right  angles.  Hence  if  NF  be  rotated  about 


Fig.  4. 

CN  until  FS  is  perpendicular  to  the  plane  CNS  (the  plane  of  the  apparent 
orbit)  right  angles  at  F  will  be  orthogonally  projected  into  the  involution  of 
conjugate  lines  at  S.  The  position  of  the  focus  F  of  the  actual  orbit  has 
therefore  been  found,  and  the  orthogonal  projection  by  which  the  true  and 
the  apparent  orbits  are  related. 

The  true  orbit  may  be  plotted  point  by  point  on  the  plane  of  the  paper, 
with  its  centre  C  and  focus  F.  For  if  P'  is  a  point  on  the  apparent  orbit  and 
P  the  corresponding  point  on  the  true  orbit  PP'  is  perpendicular  to  CN  and 
PF,  P'S  meet  on  CN.  In  particular,  if  X'  (fig.  5)  is  a  point  where  OS  meets 
the  apparent  orbit,  the  corresponding  point  X  in  which  the  perpendicular 
through  X'  to  CN  meets  CF  is  a  vertex  of  the  true  orbit  and  CX  =  a.  The 
eccentricity  is  given  by 

CS       CF 


CX'~  CX 


=  e 


Orbits  of  Double  Sf<rrx 


[CH.  x 


and  the  inclination  by 


8N 
FN 


COS  I 


where  0<t<^7r  if  the  motion  is  direct  and  \Tr<i<Tr  if  the  motion  is 
retrograde.  Also  H  (<TT)  is  the  position  angle  of  CN  and  X  is  the  angle 
between  CN  and  CF  measured  in  the  direction  in  which  the  motion  takes 
place.  The  five  geometrical  elements  of  the  orbit  have  therefore  been  found. 

100.  It  is  to  be  noticed  that  this  method  does  not  require  the  ellipse 
which  represents  the  apparent  orbit  to  be  actually  drawn.  When  the  observed 
positions  have  been  plotted  five  points  may  be  chosen  to  define  the  ellipse. 
These  points  need  not  be  actual  points  of  observation :  it  is  better  if  they  are 
graphically  interpolated  among  the  observed  positions.  Let  them  be  denoted 


90' 


Fig.  5. 

by  1,  2,  3,  4,  5.  Draw  a  line  through  1  parallel  to  23.  The  second  point  in 
which  this  line  meets  the  ellipse  can  then  be  found  by  Pascal's  theorem  with 
the  ruler  only.  This  gives  two  parallel  chords  and  hence  a  diameter. 
Similarly  a  second  diameter  is  drawn  and  the  two  intersect  in  the  centre  C 
of  the  apparent  ellipse.  Again,  by  a  similar  use  of  Pascal's  theorem,  the  points 
in  which  the  lines  IS,  2S,  3S  meet  the  ellipse  again  are  determined.  This 
gives  three  pairs  of  lines  each  of  which  determines  a  quadrangle  inscribed  in 
the  ellipse.  If  two  of  these  be  completed  the  sides  of  the  harmonic  triangles 
which  meet  in  S  determine  two  pairs  of  conjugate  lines.  From  this  point 
the  construction  follows  as  before.  The  point  X'  in  which  CS  meets  the 
apparent  ellipse  can  be  constructed  by  projective  geometry.  But  it  is 
unnecessary.  If  F'  is  the  second  focus  of  the  real  orbit  and  P  the  point 


99-ioa]  Orbits  of  Double  Stars  107 

corresponding   to   any  one    of  the  assumed    points  on  the  apparent  orbit, 
FP  +  PF'  =  2a  and  GF  =  ae.     Hence  a  and  e. 

101.  When  the  apparent   ellipse    has   been  drawn   the  eccentricity  is 
known,  for  if  CS  meets  the  ellipse  in  X',  the  projection  of  the  vertex  of  the 
true  orbit,  OS/OX '  =  e  since  the  ratio  of  segments  of  a  line  is  unaltered  by 
orthogonal   projection.     Let   GY'  be  the   conjugate  diameter  to  CX'   and 
therefore  the  projection  of  the  minor  axis  of  the  true  orbit.     If  the  oblique 
ordinates  parallel  to  GY'  are  produced  in  the  ratio  1  :  \/(l  —  e2)  an  auxiliary 
ellipse  will  be  constructed  which  is  clearly  the  projection  of  the  auxiliary 
circle  to  the  true  orbit  and  has  double  contact  with  the  apparent  orbit,  CS 
being  the  common  chord.     But  the  orthogonal  projection  of  a  circle  is  an 
ellipse  of  which  the  major  axis  is  equal  to  the  diameter  and  is  parallel  to  the 
line  of  nodes,  while  the  minor  axis  is  the  direct  projection  of  the  diameter. 
Hence  the  major  axis  of  the  auxiliary  ellipse  is  2a,  the  minor  axis  2a  cos  i, 
the  eccentricity  sin  i  and  H  is  the  angle  which  the  transverse  axis  makes 
with    the  N.  direction.     The  circle  on  the  major  axis  as  diameter  is  the 
auxiliary  circle  of  the  true  orbit  turned  into  the  plane  of  the  apparent  orbit. 
Let  X  be  the  point  in  which  this  circle  is  cut  by  a  perpendicular  from  X'  to 
the  major  axis  of  the  auxiliary  ellipse.     The  point  X  will  project  into  the 
point  X'  and  therefore  represents  the  position  of  periastron  on  the  auxiliary 
circle.    Hence  the  angle  (taken  in  the  right  sense)  which  CX  makes  with  the 
major  axis  of  the  auxiliary  ellipse,  or  line  of  nodes,  is  the  angle  X.     This  is 
the  graphical  method  of  Zwiers. 

It  is  evident  that  the  line  of  nodes  and  the  inclination  will  be  equally 
indicated  by  constructing  the  projection  of  any  circle  in  the  plane  of  the  true 
orbit.  Now  the  parameter  p  (or  semi-latus  rectum)  js  a  harmonic  mean 
between  the  segments  of  any  focal  chord.  Hence  the  circle  on  the  latus 
rectum  as  diameter  has  radii  along  any  focal  chord  which  are  equal  to  the 
harmonic  mean  of  the  focal  segments.  The  projection  of  this  circle  is  an 
ellipse  with  its  centre  at  S,  its  major  axis  equal  to  2p  and  lying  in  the 
direction  of  the  line  of  nodes,  and  its  eccentricity  equal  to  sin  i.  This  ellipse 
can  be  actually  derived  from  the  apparent  orbit  by  laying  off  on  radii  through 
S  lengths  equal  to  the  harmonic  mean  of  the  intercepts  on  the  same  chord 
between  S  and  the  curve,  since  the  ratios  are  unaltered  by  projection.  This 
principle,  of  which  another  use  will  be  made,  is  due  to  Thiele. 

102.  Such  graphical  methods  are  tedious  and  may  be  avoided  by  a  slight 
calculation  when  the  apparent  orbit  has  been  drawn.     Since  the  eccentricity 
is  known  when  this  has  been  done,  there  remain  four  geometrical  elements, 
a,  i,  n,  X,  to  be  determined.     Four  independent  quantities  are  required  and 
the  four  chosen  by  Sir  John  Herschel  and  others  are  2a,  the  diameter  through 
S,  2/3  the  conjugate  diameter,  and  ^1}  ^2  the  position  angles  of  these  diameters. 
The  length  of  the  chord  through  S  parallel  to  ft,  or  the  projection  of  the  latus 


108  Orbit*  of  Double  Stars  [en.  x 

rectum  of  the  true  orbit,  is  therefore  2/3\/(l—  e2).  Hence  the  relations 
between  the  positions  in  the  true  and  apparent  orbits  (§  98)  give  : 

a  (1  —  e)  cos  (xi  —  ft)  =  a  (1  —  e)  cos  X 
a  (1  —  e)  sin  (^  —  ft)  =  a  (1  —  e)  sin  X  cos  i 
£  V(l  -  e2)  cos  (jfc  -  ft)  =  -  a  (1  -  e*)  sin  X 
/3  V(l  —  "e2)  sin  (%2  —  ft)  =  a  (1  —  e2)  cos  X  cos  t 

since  w  =  0°  at  periastron  and  90°  at  the  extremity  of  the  latus  rectum. 
Hence  ft  is  given  by 

a2  (1  -  e2)  sin  2  (%1  -  ft)  +  /32  sin  2  (%2  -  ft)  =  0 
or 

tan  (x*  +  %2  -  2ft)  =  tan  (^  -  #2)  cos  27 
where 

tan  7  =  V(l  -  e2)  a//3. 
This  equation  in  ft  is  satisfied  by  ft  +  \TT  as  well  as  ft.     But 

cos2  i  =  —  tan  (  ^  —  ft)  tan  (^;2  —  ft) 

and  this  rejects  ft  +  £TT  since  cos  i  <  1  and  determines  i.  The  first  and  third 
of  the  above  set  of  four  equations  give  both  a  and  X  with  its  proper  quadrant 
and  the  second  or  fourth  gives  also  the  proper  sign  of  cos  i  (according  to  the 
convention  of  §  98).  The  solution  is  then  free  from  ambiguity,  understanding 
that  ^j  is  the  position  angle  corresponding  to  periastron  and  ^2  the  position 
angle  when  the  companion  has  moved  through  one  quadrant  in  its  plane 
beyond  this  point. 

103.     Another  method  employs  the  general  equation 
,  owr2  +  Zhxy  +  by2  +  2gx  +  2fy  +  c  =  0 

of  the  apparent  orbit  referred  to  the  principal  star  as  origin.  Without  loss  of 
generality  c  may  be  put  equal  to  1.  The  other  coefficients  are  to  be  chosen 
to  satisfy  the  observations  as  well  as  may  be.  But  an  elaborate  solution  is 
not  justified  because  the  one  accurate  element  in  the  observation,  the  time, 
is  not  involved  in  this  stage.  The  intersections  of  the  ellipse  with  the  axes 
and  any  fifth  point  give  the  result  in  the  simplest  way.  The  elements  of  the 
true  orbit  can  then  be  derived  in  a  variety  of  forms.  Let  us  find  the  pro- 
jection of  the  circle  on  the  latus  rectum.  The  above  equation  may  be  written 

2  c 

a  cos2  6  +  2k  cos  9  sin  6  +  b  sin2  9  +  -  (g  cos  6  +/sin  6}  +  —  =  0. 

For  a  particular  value  of  B,  p  has  two  values,  pl  and  —  p2,  one  positive  and 
one  negative  since  the  origin  is  inside  the  curve.  Hence,  if  p  represents  the 
harmonic  mean, 

1      1/1       IN2      1/1       IN2       1 

-2  =  T-+~      =A(  ---  ~ 

p-     4  V/3i     pj       4  V/3i      pj 


=  {(g  cos  0  +/sin  0)*  -c(a  cos2  6  +  2h  cos  0  sin  0  +  b  sin2  0)}/c2 
=  (-  B  cos2  e  +  2H  sin  6  cos  6  -  A  sin2  6)1  c- 


102,  los]  Orbits  of  Double  Stars  109 

where,  in  the  usual  notation, 

A  =  be  -f",     H  =fg  -ch,     B  =  ac  -  g\ 

Hence  the  equation 

Bx>  -  2Hx;y  +  Ay2  +  c2  =  0 

represents  the  projection  of  the  circle  on  the  latus  rectum  (§  101),  or  an 
ellipse  with  axes  2p  and  2p  cos  i  and  its  transverse  axis  coinciding  with  the 
line  of  nodes.  It  is  therefore  identical  with  the  equation 

(x  cos  ft  4-  y  sin  ft)2     (y  cos  ft  —  x  sin  ft)2  _ 

p2  p"  cos2  i 

and  thus 

-  B/c2  =  p~2  cos2  ft  +  p~*  sec2  i  sin2  ft 

H/c*  =  (  p~*  —  p~z  sec2  i)  sin  ft  cos  ft 

-  ^1/c2  =  ^>-2  sin2  ft  +  p~z  sec2  1  cos2  ft 
or 

p-2  tan2  1  sin  2ft  =  -  2#/c2 

~2  tan2  i  cos  2ft  =  5  - 


2p~2  +  p~2  tan2  1  =  -  (B  +  A)/c2 
which  determine  ft,  p  and  i. 

Again,  the  perpendicular  from  the  focus  on  the  directrix  is  a  (e~l  —  e)  =  pe~*. 
Hence  the  intercepts  on  the  line  of  nodes  and  on  the  line  perpendicular  to  it 
between  the  focus  and  the  directrix  are  p/e  cos  X,  p/e  sin  A,.  The  projections 
of  these  intercepts,  also  at  right  angles,  are  p/e  cos  X,  p  cos  i/e  sin  X.  But  the 
projection  of  the  directrix  is  the  polar  of  the  origin,  or  the  line  gx  +fy  +  c  =  0.  . 
Hence 

(g  cos  fl  +/sin  H)  p/e  cos  X  +  c  =  0 

(—  g  sin  n  +/cos  fl)  p  cos  i/e  sin  X  +  c  =  0 
so  that  e  and  X  are  given  by  the  equations 

e  sin  \  =  —p  cos  i  (/cos  H  —  g  sin  ft)/c 
e  cos  X  =          —  p  (/sin  fl  +$r  cos  fl)/c. 

Equations  for  the  five  geometrical  elements  in  the  above  form  were  first  given 
by  Kowalsky. 

The  form  of  the  equation  which  represents  the  projection  of  a  circle  is 
defined  by  the  fact  that  the  asymptotes  of  the  projected  ellipse  are  parallel 
to  the  projection  of  the  circular  lines  and  therefore  to  the  tangents  from  S  to 
the  apparent  orbit.  It  will  be  found  that  the  projection  of  the  auxiliary 
circle,  referred  to  its  centre,  is  in  the  usual  notation 

C'2  (Bx-  -  ZHxy  +  Ay-)  +  A2  =  0 


110  Orbits  of  Double  Star*  [CH.  x 

and  that  of  the  director  circle 

C* (Bx>  -  ZHxy  +  Ay2)  +  A  (A  +  Cc)  =  0 
while  the  eccentricity  of  the  true  orbit  is  given  by 

l-e*  =  G'c/A. 

104.  In  some  few  cases  a  double  star  has  been  observed  over  more  than 
one  complete  revolution.  The  period  P  is  then  known  approximately  and 
the  date  T  of  periastron  passage,  when  the  companion  is  situated  on  the 
diameter  of  the  apparent  orbit  through  8.  Otherwise,  when  the  geometrical 
elements  have  been  determined,  two  dated  observations  suffice  to  determine 
these  two  additional  elements.  For  two  observed  position  angles  6l,  02  give 
the  corresponding  true  anomalies  wlt  w.2  and  hence  the  eccentric  anomalies 

EI,  E2,  since 

/n  —  e\ 
tan  (6  -  H)  =  tan  (w  +  \)  cos  i,      tan  %E  —        L-          tan  ^w. 

v     \1  ~t~  6/ 
Then 

n  (t,  -T)  =  El-e  sin  Elt     n  (t,  -T)  =  E2-e  sin  E* 

determine  n  =  2-7T/P  and  T.  In  practice  a  larger  number  of  such  equations 
will  be  employed  in  order  to  reduce  the  effect  of  errors  in  the  observations. 
The  law  of  areas  can  also  be  applied  directly  to  the  apparent  orbit,  for  if  a^ 
is  the  area  described  by  the  radius  vector  between  the  dates  tlt  t.z,  and  Al  is 
the  area  of  the  ellipse,  P  =  (£2  —  O  ^i/ff-i,  and  similarly  T  can  be  determined. 
A  primitive  method  which  has  been  used  for  measuring  the  areas  consists  in 
cutting  out  the  areas  in  cardboard  and  weighing  them. 

When  the  parallax  w  of  a  double  star  is  known,  a/vr  is  the  mean  distance 
in  the  system  expressed  in  terms  of  the  astronomical  unit.  Hence  (§  24),  if 
m,  in'  are  the  masses  of  the  components, 

k*  (m  +  m')  =  47rsa8/w3VJ8 

while  A;2  =  4-Tr2  if  the  mass  of  the  Sun-Earth  system  and  the  sidereal  year  are 
taken  as  units.  For  this  purpose  the  mass  of  the  Earth  is  negligible  and 
thus,  P  being  expressed  in  years, 

,m  -f  m  =  a?/tv3P- 
is  the  combined  mass  of  the  system,  compared  with  that  of  the  Sun. 

105.  The  apparent  orbit  can  be  reconstructed,  on  an  arbitrary  scale, 
from  observed  position  angles  alone.  This  course  was  advocated  by  Sir  J. 
Herschel,  who  considered  the  measured  distances  of  his  day  very  inferior  in 
accuracy.  With  this  object  the  position  angles  are  plotted  as  ordinates  with 
the  time  as  abscissa.  Owing  to  inaccuracies  the  points  will  not  lie  exactly 
on  a  smooth  curve,  but  such  a  curve  must  be  drawn  through  them  as  well  as 
possible.  Let  i/r  be  the  angle  which  the  tangent  to  the  curve  at  the  point 


103-105]  Orbits  of  Double  Stars  111 

(t,  6}  makes  with  the  axis  of  ty  so  that  d0/dt  =  tan  ty.  But  since  Kepler's 
law  of  areas  is  preserved  in  the  apparent  orbit,  p26  =  h,  an  undetermined 
constant.  Hence  p  =  \/(h  cot  \|r)  and  the  apparent  orbit  can  thus  be  derived 
graphically  from  the  (t,  6}  curve.  The  elements  with  the  exception  of  a  can 
then  be  obtained  and  finally  a  is  determined  by  the  measured  distances,  of 
which  no  other  use  is  made  in  the  calculation. 

The  opposite  case  may  arise,  and  is  illustrated  by  the  star  42  Comae 
Berenices,  in  which  the  determination  of  the  elements  must  be  based  on  the 
distances.  Here  the  plane  of  the  orbib  passes  through  the  point  of  observa- 
tion, i  =  90°  (or  practically  so)  and  the  position  angles  serve  only  to  determine 
O.  If  the  star  has  been  observed  over  more  than  one  revolution  the  period  P 
may  be  considered  known.  Corresponding  to  the  point  (a  cos  E,  b  sin  E)  on 
the  orbit,  the  observed  distance  is 


while 


p  =  a  cos  E  cos  X  —  b  sin  E  sin  X  -  ae  cos  X 
=  R  cos  (E  +  /3)  —  ae  cos  X 


If  the  observations  are  plotted  for  a  single  period,  from  maximum  to 
maximum,  the  result  is  to  give  the  curve 

a;  =  nt  =  nT+  E  —  e  sin  E 

y  =  p  =  R  cos  (E  +  /3)  —  ae  cos  X 

which  is  a  distorted  cosine  curve.     Maximum  and  minimum  correspond  to 

E  =  —  /3,  TT  —  /3  and  give 

nti  =  nT  —  /3  +  e  sin  /3,  y^  =  R  —  ae  cos  X 

nt2  —  nT  +  TT  —  /3  —  e  sin  /3,     y2  =  —  R  —  ae  cos  X 
whence  R  and  ae  cos  X,  while  in  addition 

n  (t2  -  ti)  =  TT  —  2e  sin  /3. 

These  equations  may  be  supplemented  by  a  simple  device.  Taking  the 
origin  of  a;  at  the  first  maximum  let  the  curve 

y  —  R  cos  x  —  ae  cos  X 

also  be  drawn.  Let  P  be  a  point  on  this  curve  and  Q  the  corresponding 
point  on  the  first  curve  such  that  the  ordinates  at  P  and  Q  are  equal.  Then 
at  P,  x  =  E  +  (3,  so  that 


Hence  the  curve 

y  =  esin(x  —  {3)  +  ft  —  n(T  —  £,) 

can  be  constructed  by  laying  off  on  each  ordinate  through  P  a  length  equal 
to  QP.     This  is  a  simple  sine  curve,  the  form  of  which  will  serve  to  show 


112  Orbits  of  Double  Star*  [CH.  x 

any  irregularities  in  the  (nt,  p)  curve  from  which  it  is  derived.  The  ampli- 
tude is  2e,  represented  on  the  scale  by  which  2?r  corresponds  to  the  period  in 
x.  The  value  of  e  being  thus  known  gives  /3  from  (t.2  —  £x)  and  hence  a  and  X, 
since 

a  cos  \  =  R  cos  /3,     a  sin  \  =  R  sin  /3/\/(l  —  e2). 

T  is  then  given  by  the  maximum  and  minimum  of  the  original  curve.  But 
the  sine  curve  has  its  maximum  at  x  =  /3  +  |TT  and  its  central  line  is 
y  =  ft  —  n(T  —  tj).  These  conditions  must  also  be  fairly  satisfied  by  the 
adopted  solution. 

106.  Graphical  methods,  such  as  those  sketched  above,  only  provide  a 
first  approximation  to  the  solution  of  a  problem.  Here  in  general  the  obser- 
vations are  too  rough  to  make  a  closer  approximation  feasible.  But  if  it  is 
necessary  to  improve  the  elements  thus  found,  each  observation  gives  one 
equation  in  the  following  way.  Let  da,  dfl,  ...  be  the  required  corrections 
to  the  approximate  elements,  a,  H,  ....  For  the  time  t  of  an  observation 
6  (or  p)  can  be  calculated.  Its  value  is 

ec=f(t,a,  n,  ...). 

But  the  observed  value  is 

eo  =f(t,  a  +  da,  n  +  dfl,  .  .  .). 

If  then  the  elements  have  been  found  with  such  an  accuracy  that  squares, 
products  and  higher  powers  of  da,  dfl,  .  .  .  can  be  neglected, 

60  -  6C  =  ^~  .  da  +  ^.  .  d£l  +  .  .  . 

oa  oil 

a  linear  equation  in  da,  d£l,  ....     And  similarly  with  p.     The  coefficients  are 

—  =  0  ?£  =  £ 

da  da     a 


-  - 

an  an~ 

.~r  =  —  ^  sin  2  (6  —  H)  tan  i,  ^.  =  -p  sin-  (6  -  H)  tan  i 

^-  =  —  cos  i,  £  =  —  £p  sin  2  (6  —  H)  sin  i  tan  i 

o\     p-  d\ 

80         no*        .  dp         na2  (       .     „  .  dp 

^=  -  —  cos  i  V(l  -  e*),  ^T=   -  —  pepsin  E  +  V(l  -  e'-) 


__ 
dn  n    'dT'  dn~         n     '  dT 

d0      r"  fa         1     \    .  .        dp      dp  fa          1     \  ap 

5-  •*  -;  I  -  +  1       ,   sm  w  cos  i,       ^  =  £  (-  +  -       -    sin  w  —  -  cos  w 

de      p-\r      \-e-J  de     d\\r      1  -  e-J  r 

• 

the  verification  of  which  may  be  left  as  an  exercise. 


105-108]  Orbits  of  Double  Stars  113 

107.  In  some  cases  the  position  of  a  binary  system  has  been  measured 
relatively  to  some  neighbouring  star  C  which  is  independent  of  the  system. 
Let  A  be  the  principal  star,  m^  its  mass,  (acl}  y^  its  coordinates  at  the  time  t', 
and  similarly  let  B  be  the  companion,  m2  its.  mass,  (xz,  y2)  its  coordinates. 
A  series  of  measures  of  AB  gives 

x.2  —  x1  =  p  cos  0,    2/2  —  2/i  —  P  sm  $ 

while  the  measures  of  AC  give  x3  —  xl}  y3  —  yi,  (#:i,  y3)  being  the  position  of  C. 
Let  (f,  77)  be  the  c.G.  of  AB,  so  that 

(mi  +  m2)  £  =  mlxl  +  m2x2,     (r^  +  w2)  <>7  =  m^  +  7n2i/2. 

But  the  motions  of  C  and  of  the  C.G.  of  AB  are  uniform  and  independent. 
Hence 

|  =  x3  +  a  +  fit,     i]  =  2/3  +  a'  +  fit 

where  ft,  ft'  are  the  proper  motions  of  the  C.G.  relative  to  C,  and  (a,  a')  is  its 
position  relative  to  C  at  the  chosen  epoch  to  which  t  refers.  Thus 

(ml  +  m.2)  (xs  +  a  +  fit)  =  mlxl  +  m^xz 
or 

a  +  fit  —f(ccz  —  #j)  +  %3  —  xl  =  0 
and 

a'  +  fft  -f(y,  -y1)  +  y3-y1  =  0 
similarly,  where 

/=  TWa/CWj  +  ma). 

From  a  series  of  such  equations  a,  a',  /3,  /3'  and  f  can  be  determined  and 
therefore  the  ratio  of  the  masses  of  A  and  B.  But  if  a  is  the  mean  distance, 
P  the  period  and  to-  the  parallax  of  the  system  AB, 


and  the  masses  of  the  individual  stars,  expressed  in  terms  of  the  Sun,  become 
known. 

108.  In  certain  cases  the  absolute  coordinates  of  stars  apparently  single 
have  exhibited  a  variable  proper  motion.  It  is  then  assumed  that  the  varia- 
tion is  periodic  and  due  to  orbital  motion  in  conjunction  with  an  undetected 
body.  The  motion  to  be  investigated  is  relative  to  the  C.G.  of  the  system, 
which  itself  is  supposed  to  move  uniformly.  In  the  plane  of  the  orbit  the 
coordinates  are  a'  (cos  E  —  e),  b'  sin  E,  and  therefore  in  the  plane  of  projection, 
when  referred  to  the  line  of  nodes  and  the  line  at  right  angles,  they  become 

x  =  a'  (cos  E  —  e)  cos  A,  —  b'  sin  E  sin  \ 
y={af  (cos  E  —  e)  sin  X  +  b'  sin  E  cos  X}  cos  *'. 

Hence  the  orbital  displacement  in  the  direction  of  the  position  angle  Q  is 
£  =  ayos  (tl-Q)-y  sin  (ft  -  Q) 

=  g  cos  E  +  h  sin  E  —  ge 
p.  D.  A.  8 


114  Orbits  of  Double  Stars  [CH.  x 

where 

g  —     a  {cos  X  cos  (11  —  Q)  —  sin  X  sin  (O  —  Q)  cos  i] 

h  =  —  b'  {sin  A,  cos  (O  —  Q)  +  cos  A,  sin  (H  —  Q)  cos  i] 

and  Q  =  90°  for  displacements  in  R.A.,  Q  =  0°  for  displacements  in  declination. 
The  observations  of  one  coordinate,  say  8,  therefore  give  a  series  of  equations 
of  the  form 

S  =  B0  +  fj>&t  +  g  cos  E  +  h  sin  E  —  ge 
with 

E  -  e  sin  E  =  n  (t  -  T). 

From  these  e,  n  (or  P),  T,  /AS,  &0>  g  and  h  can  be  determined.  Since  g  and  h 
are  functions  of  a,  H,  A,  and  i,  these  four  elements  cannot  be  derived  from 
observations  of  one  coordinate  alone.  But  from  observations  of  the  other 
coordinate,  say  a,  the  corresponding  quantities  g  and  h'  can  be  found  and  the 
elements  of  the  motion  are  then  completely  determinate,  including  a',  the 
mean  distance  from  the  C.G.  of  the  system. 

In  the  two  notable  examples  of  this  kind,  Sirius  and  Procyon,  the 
companion  was  discovered  afterwards.  It  thus  became  possible  to  find  the 
relative  mean  distance  a  and  hence  the  ratio  of  the  masses,  since 


Hence,  the  parallax  being  known,  the  individual  masses  of  the  components 
have  been  determined.  It  is  to  be  noticed  that,  when  the  companion  cannot 
be  observed,  the  function  of  the  masses  which  can  be  found  is  mj'  (n^  +  m.2) 
For  this  is  equal  to  of'/v^P*. 


~2 


CHAPTEE  XI 

ORBITS    OF   SPECTROSCOPIC    BINARIES 

109.  Another  class  of  orbits  which  are  based  on  pure  elliptic  motion  is 
presented  by  those  systems  which  are  known  as  spectroscopic  binaries.  It 
is  now  possible  to  determine  the  radial  velocities  of  the  stars  in  absolute 
measure  with  high  accuracy.  This  follows  from  the  application  of  Doppler's 
principle  to  the  interpretation  of  stellar  spectra.  On  the  simple  wave  theory 
of  light  this  principle  is  easily  explained.  A  light  disturbance  travels  out- 
wards from  its  source  in  a  spherical  wave  front  which  expands  in  the  free 
ether  of  space  with  the  uniform  velocity  U.  Let  a  fixed  set  of  rectangular 
axes  be  taken  in  this  space,  and  let  (xl,  ylf  Zj)  be  the  position  of  the  source 
at  the  origin  of  time.  Let  (ulf  vlt  w^  be  the  velocity  components  of  the 
source,  supposed  to  be  in  uniform  motion,  and  t  the  time  at  which  a  light 
disturbance  is  emitted.  Similarly  let  (#2>  2fe>  ^2)  be  the  position  of  the 
observer,  also  supposed  to  be  moving  uniformly,  («2,  v2,  iu2)  the  velocity 
components,  and  r  the  time  at  which  the  specified  disturbance  reaches  him. 
For  simplicity  the  motions  have  been  considered  uniform,  but  obviously  they 
are  immaterial  except  as  regards  the  source  at  the  instant  t  and  the  observer 
at  the  instant  T.  Let  the  corresponding  positions  be  A,  B  respectively  and 
let  the  distance  AB  =  R.  Then 


dR      _-,    /     dr         \      tr  dr 


where  (a,  ft,  7)  are  the  direction  cosines  of  AB  and  Vlt  F2  are  the  projections 
of  the  velocities  (uv,  vlt  w^),  (u2,  v2,  w2)  on  this  line.  But  since  the  wave 
reaches  B  from  A  in  the  time  (r  —  t), 


Hence 


,_    ,    .->  , 


dt      U-V.,  U          U(U-VZ)  ' 

8—2 


116  Orbits  of  Spectroseopic  Binaries  [CH.  xi 

Now  (  V2  —  Vi)  is  the  component  of  relative  velocity  of  A  and  B,  measured 
in  the  direction  of  separation  of  the  two  points.  This  is  a  definite  quantity. 
But  F2  is  a  component  of  the  observer's  absolute  motion  in  free  ether,  and 
this  is  unknown.  Presumably  it  is  small  in  comparison  with  U,  and  the  last 
term  can  be  rejected  as  a  negligible  effect  of  the  second  order.  Or,  on  the 
theory  of  relativity,  Vz  is  not  only  unknown  but  unknowable,  and  the  effect 
is  completely  compensated  by  a  transformation  of  the  ideal  coordinates  of 
space  and  time  into  another  set  which  is  the  subject  of  observation.  All 
this  has  its  counterpart  in  the  theory  of  aberration,  with  which  it  is  intimately 
related.  Whether  the  limitation  is  imposed  by  the  imperfection  of  practical 
observations  or  by  the  ultimate  nature  of  things,  it  is  necessary  to  be  content 
with  the  effect  of  the  first  order. 

If  the  light  emitted  at  A  has  the  wave  length  \,  the  frequency  of  a 
particular  phase  in  the  wave  train  at  A  is  U/\.  But  the  number  of  waves 
emitted  in  a  time  dt  is  received  at  B  in  the  time  dr.  If  then  the  apparent 
wave  length  of  the  light  received  at  B  is  X'  and  the  apparent  frequency 

U/\', 

UX'1  dt  =  UX'-1  dr 
and  therefore 

V_rfr_         V 

\  ~  dt  ~     f  U 

where  V  is  the  relative  radial  velocity  of  A  from  B.  Thus  the  application 
of  Doppler's  principle  gives 


where  AX  is  the  increase  of  wave  length  (or  displacement  measured  positively 
towards  the  red  end  of  the  spectrum)  of  a  spectral  line,  of  which  the  natural 
'wave  length  in  the  star  is  supposed  known.  Further  details  on  the  practical 
methods  of  reduction  would  be  out  of  place  here,  and  this  explanation  must 
suffice.  It  is  usual  to  express  V  in  km.  /sec.,  and  the  velocity  of  light  maybe 
taken  to  be  U=  299860  km.  /sec. 

110.  From  the  measured  radial  velocity  must  be  deduced  the  radial 
velocity  of  the  star  relative  to  the  Sun,  or  rather  relative  to  the  centre 
of  gravity  of  the  solar  system.  This  requires  the  calculation  of  certain 
corrections,  of  which  the  most  important  are  due  to  (1)  the  diurnal  rotation 
of  the  observer,  and  (2)  the  annual  elliptic  motion  of  the  Earth  relative  to 
the  Sun.  The  effects  of  perturbations  of  the  Earth  and  Sun  are  compara- 
tively small. 

An  observer  situated  on  the  equator  is  carried  by  the  Earth's  rotation 
over  40,000  km.  in  a  sidereal  day.  This  means  a  velocity  of  0'46  km.  /sec. 
Hence  the  velocity  of  an  observer  in  latitude  <£  is  0'46  cos</>  km.  /sec.  always 
directed  towards  the  E.  point.  If  0  is  the  angular  distance  of  the  star  from 
this  point  at  the  time  of  observation,  cos  9  =  cos  8  cos  (h  +  90°),  where  B  is  the 


109,  no]  Orbits  of  Spectroscoptc  Binaries  117 

declination  and  //  the  W.  hour  angle  of  the  star.  Hence  the  additive 
correction  corresponding  to  (1)  is 

vd  =  +  0-46  cos  <£  cos  Q--  0*46  cos  <f>  cos  8  sin  h. 

Again,  the  Earth's  elliptic  velocity  is  compounded  (§  26)  of  one  constant 
velocity  Vl  perpendicular  to  the  radius  vector  and  another  eVl  perpendicular 
to  the  major  axis,  e  being  the  eccentricity  of  the  orbit.  These  .vectors  are 
directed  to  points  in  the  ecliptic  of  which  the  longitudes  are  ©  —  90°  and 
F  —  90°,  where  8  is  the  longitude  of  the  Sun  and  F  the  longitude  of  the  • 
solar  perigee.  Let  (I,  /3)  be  the  star's  longitude  and  latitude.  Hence  the 
required  correction  for  the  Earth's  orbital  motion  is 

Va  =  +Vl  cos  /3  [cos  (I  -  B  +  90°)  +  e  cos  (1-T  +  90°)}. 

Now  Fj  is  precisely  that  vector  on  which  the  constant  of  stellar  aberration 
depends,  so  that  if  k"  is  this  constant, 

V1  =  k"Uj 206265"  =  29-76  km./sec. 

when  the  standard  value  of  k,  20"'47,  is  adopted  with  the  value  of  U  given 
above.  Hence  the  correction  for  (2)  is 

va  =  +  2976  cos  £  (sin  (©  - 1)  +  e  sin  (T  -  I)}. 

It  is  evident  that  the  process  might  be  reversed  and  the  value  of  k  deter- 
mined by  observing  the  apparent  radial  motion  of  one  or  more  stars  at 
different  times  of  year.  This  has  been  done  at  the  Cape  Observatory,  with 
the  result  that  the  standard  value  of  k  was  reproduced  very  exactly,  an 
excellent  test  of  the  theory.  Indeed  this  is  probably  the  best  available 
method  of  finding  the  constant  of  aberration :  it  will  be  noticed  that  the 
adopted  value  of  U,  being  a  factor  of  both  Vl  and  V,  will  scarcely  affect  the 
resulting  value  of  k. 

When  the  necessary  corrections  have  been  applied  to  the  apparent  radial 
velocity  of  a  star,  the  star's  radial  velocity  is  obtained  relative  to  the  solar 
system.  This  is  affected  by  the  motion  of  the  latter  relative  to  the  stellar 
system  as  a  whole.  Hence  conversely  when  the  radial  velocities  of  a  number 
of  stars  scattered  over  the  sky  are  known,  it  becomes  possible  to  deduce  the 
motion  of  the  solar  system  relative  to  the  average  of  those  stars  in  absolute 
measure.  If,  further,  CT  is  the  parallax  of  a  star,  and  /*  its  total  annual 
proper  motion,  its  transverse  velocity  is  p/nr  when  expressed  in  astronomical 
units  per  year.  Now  with  the  solar  parallax  8"'80  and  the  Earth's  equatorial 
radius  6378'249  km.,  the  astronomical  unit  (or  Earth's  mean  distance  from 
the  Sun)  is  149,500,000  km.  Hence  this  unit  of  velocity  is  equivalent  to 
4*737  km./sec.  and  the  star's  transverse  velocity  is  4737  /H/OT  km./sec.  Thus 
the  velocity  of  a  star  relative  to  the  Sun  can  be  completely  determined  in 
absolute  measure.  This  concerns  questions  of  stellar  kinematics  which  are 
now  entering  the  region  of  dynamics  but  lie  outside  our  present  scope. 


118  Orbits  of  Spectroscopic  Binaries  [CH.  xi 

111.  Repeated  determinations  of  the  radial  velocity  of  a  star  yield  values 
which  in  the  majority  of  cases  are  constant  within  the  errors  of  observation. 
The  motion  of  the  star  is  apparently  uniform.     But  in  other  cases,  perhaps 
a  third  of  all  the  brighter  stars,  changes  are  observed  which  prove  to  be 
regular  and  periodic.     These  are  attributed  plausibly  to  the  motion  of  one 
component   in   a   binary  system.     Such    spectroscopic   binaries   differ  from 
visual  doubles  only  in  the  scale  of  their  orbits,  which  prevents  them  from 
being  resolved  even  in  the  most  powerful  telescopes,  while  their  periods  are 
to  be  reckoned  in  days  instead  of  years  or  even  centuries.     It  may  appear 
that  the  spectrum  of  the  second  component  should  also  be  seen.     When  the 
components  are  fairly  equal  in  brightness,  as  in  ft  Aurigae,  this  is  so ;  the 
lines  of  the  spectrum  are  seen  periodically  doubled.     But  with  other  stars, 
and  this  is  the  more  common  type,  the  companion  is  relatively  so  faint  that 
only  one  spectrum  is  shown :   it  is  quite  unnecessary  to  suppose  that  the 
companion  is  then  an  absolutely  dark  body.     Even  when  both  spectra  are 
visible  the  secondary  spectrum  is  often  difficult  to  detect  and  usually  difficult 
to  measure.     As  a  particularly  interesting  example  Castor  (a  Geminorum) 
may  be  quoted.     The  telescope  reveals  this  star  as  a  visual  double,  and  the 
spectroscope   shows  that  both   components  are   themselves  binary  systems. 
More  complex  systems  can  be  inferred  from  spectroscopic  measures  alone. 
Thus  Polaris,  which  appears  in  the  telescope  as  a  single  star,  has  been  shown 
to  be  a  triple  system,  consisting  of  a  close  pair  revolving   round  a  more 
distant  third  body.     Here  the  motion  will  be  considered  in  the  first  instance 
of  one  component  of  a  binary  system  about  the  common  centre  of  gravity, 
and  it  will  be  seen  how  far  the  elements  of  an  elliptic  orbit  can  be  deduced 
from  the  measured  radial  velocities,  these  being  based  on  the  comparison  of 
the  star's  spectrum  with  that  from  a  terrestrial  source  (usually  the  spark 
spectrum  of  iron  or  titanium). 

112.  Since  the  period  is  generally  short,  the  observations  extend  over 
several  revolutions  and  the  period  P  is  determined  by  obvious  considerations 
with   fair   exactness.     This   being   known,  the   observed    velocities   can  be 
referred  to  a  single  period  with  arbitrary  epoch  and  plotted  as  ordinates 
with  the  time  as  abscissa  in  a  diagram  called  the  radial  velocity  curve.     Such 
a  curve  is  illustrated  in  fig.  «,  while  the  relative  orbit  is  shown  in  fig.  b, 
corresponding  points  being  indicated  by  the  same  letters.     The  focus  of  this 
orbit  is  G,  the  centre  of  gravity  of  the  system.     The  line  of  nodes  AGB, 
passing  through  A  the  receding  node  and  B  the  approaching  node,  is  the 
line  drawn  through  G  in  the  plane  of  the  orbit  at  right  angles  to  the  line  of 
sight.     The  points  P1}  P2  mark  the  position  of  periastron  and  apastron,  and 
the  angle  from  G  A  to  GP^ ,  measured  in  the  direction  of  motion,  is  the  longi- 
tude of  periastron,  w.     The  true  anomaly  at  any  point  of  the  orbit  being  iv, 
the  longitude  of  this  point  from  A  is  u  =  o>  +  w.     Let  i-  (0°  <  i  <  90°)  be  the 


Ill,  112] 


Orbits  of  Spectroscopic  Binaries 


119 


inclination  of  the  orbit,  this  being  the  angle  between  its  plane  and  the  plane 
which  is  normal  to  the  line  of  sight,  and  let  e  be  the  eccentricity. 


150  200  250  300  days 


Fig.  6 :  (a)  upper,  (b)  lower. 

The  orbital  velocity  of  the  star  is  compounded  (§  26)  of  one  constant 
velocity  V2  transverse  to  the  radius  vector  and  another  eV2  perpendicular  to 
the  major  axis.  These  may  be  resolved  along  and  perpendicular  to  the  line 
of  nodes.  The  former  components  contribute  nothing  to  the  radial  velocity. 
The  latter  are  +  V2  cos  u  and  +  e  F2  cos  &>  in  the  direction  GE  which  is 


120  Orbits  of  Spectroscopic  Binaries  [CH.  xi 

drawn  at  right  angles  to  GA.  This  line  makes  the  angle  (90°  -  i)  with  the 
line  of  sight,  and  hence  the  radial  velocity  which  is  measured  is 

V  =  7  4-  (cos  u  +  e  cos  &>)  V2  sin  i 

where  7  is  the  radial  velocity  of  the  point  G,  that  is,  of  the  system  relative 
to  the  Sun.  It  is  at  once  evident  that  V2  and  i  cannot  be  determined  inde- 
pendently from  the  radial  velocities  alone,  and  the  equation  may  be  written 

V—  7  +  K(cos  u  +  ecosco),     K=Vzsini 
or  again, 

V  =  7'  +  K  cos  u,     7'  =  7+  Ke  cos  &> 

where  K,  7  and  7'  are  to  be  taken  as  constant. 

113.  When  the  velocity  curve  has  been  drawn  the  maximum  and  mini- 
mum ordinates  are  approximately  known.  These  are  y  =  7'  +  K,  y  =  7'  —  K, 
which  require  u  =  0,  u  =  180°.  The  maximum  and  minimum  points,  A,  B, 
therefore  correspond  with  the  receding  and  approaching  nodes.  The  line 
y  =  y  can  then  be  drawn  in  the  diagram,  intersecting  the  velocity  curve  in 
E,  F.  These  points  require  u  =  90°,  270°  and  the  corresponding  points  in 
the  orbit  are  the  extremities  of  the  focal  chord  at  right  angles  to  the  line  of 
nodes.  The  velocity  curve  is  thus  divided  at  A,  E,  B,  F  into  four  parts 
corresponding  to  four  focal  quadrants,  each  bounded  on  one  side  by  the  line 
of  nodes.  The  part  which  contains  the  periastron  passage  will  be  described 
in  the  shortest  time  and  that  which  contains  the  apastron  passage  will 
require  the  longest  time.  The  opposite  extremities  of  any  focal  chord  give 
equal  and  opposite  values  to  (V—y)-  In  particular,  the  periastron  and 
apastron  points,  P1}  P2,  are  located  on  the  velocity  curve  by  the  further 
condition  that  their  abscissae  differ  by  £P,  the  half  period,  and  the  points 
LI,  L2  corresponding  to  the  ends  of  the  latus  rectum  by  the  condition  that 
they  are  equidistant  in  time  from  Pl  or  P2.  The  four  points  Plf  P2,  L1}  L2 
on  the  velocity  curve  are  easily  found  graphically  by  trial  and  error. 

Again,  let  0  be  the  centre  of  the  orbit  and  COD  the  diameter  which  is 
conjugate  to  the  diameter  parallel  to  the  line  of  nodes,  so  that  the  tangents 
to  the  orbit  at  C  and  D  are  also  parallel  to  this  line.  Hence  V=y  at 
C  and  D  on  the  velocity  curve.  Let  an  axis  of  z  be  taken  parallel  to  GE  in 
the  plane  of  the  orbit,  so  that 

T7  dz   . 


t, 

Now  the  integral  represents  the  area  of  the  velocity  curve  measured  from 
the  line  y  =  7.  Hence  by  taking  the  limits  at  A,  C,  B,  D  it  follows  that  the 
positive  area  of  the  velocity  curve  from  A  to  C  is  equal  to  the  negative  area 
from  C  to  B,  and  the  negative  area  from  B  to  D  is  equal  to  the  positive  area 


112-H4]  Orbits  of  Spectroscopic  Binaries  121 

from  D  to  A.    These  conditions,  which  can  be  tested  by  a  planimeter  or  some 
equivalent  method,  make  it  possible  to  draw  the  line  y  —  7  in  the  diagram. 

At  Klf  K2,  the  extremities  of  the  minor  axis,  the  radial  velocities  relative 
to  G  are  equal  and  opposite.  Hence  on  the  velocity  curve  K^  and  K2  are  at 
equal  and  opposite  distances  from  the  line  y  =  7  and  equidistant  in  time 
from  P1  or  P2.  Thus  these  points  can  also  be  found  graphically  without 
difficulty. 

114.  It  is  supposed  that  the  period  P  is  known,  and  this  gives  the  mean 
daily  motion,  yu,=  27r/P.  The  other  quantities  which  can  be  derived  from 
the  velocity  curve  are  five  in  number,  namely  T  the  time  of  periastron 
passage,  K  =  V2  sin  i,  y  the  radial  velocity  of  the  system,  &>  the  longitude  of 
the  node,  and  e  =  sin  0  the  eccentricity  of  the  orbit.  The  most  satisfactory 
direct  method  of  finding  these  elements  is  based  on  the  representation  of 
the  curve  (see  Chapter  XXIV)  by  a  harmonic  series  in  the  form 

V=  Fo  +  2r,-sinO>*  +  $) 

where  t  is  reckoned  from  some  arbitrary  epoch.     This  is  always  possible 
by  Fourier's  theorem.     But 

V  =  7  +  K  cos  G)  (e  +  cos  w)—K  sin  &>  sin  w 
=  7  +  *2K  cos  «D  cos2  <f>  .  e~l  2  J,-  (je)  cosjM 
—  2K  sin  w  cos  <J>  .  ^J/  (je)  sinjM 

by  §  41,  (28)  and  (29).     Now  M  =  p(t-T)  and  therefore  F0  =  7  and 
rj  sin  (jfiT  +  ft)  =  2K,  .  er*  Jj  (je) 


where 

K^  =  K  cos  to  cos2  </>,     K2  =  K  sin  &>  cos  <f>    ............  (1) 

There  are  now  only  four  quantities  to  be  determined,  which  may  be  taken  to 
be  K-i,  K2,  T  and  e.  Thus  the  four  equations  corresponding  to  j  =  1,  2  are 
alone  required  :  those  of  a  higher  order  are  useful  only  when  there  is  reason 
to  suspect  that  the  motion  is  not  purely  elliptic.  Now  these  give  (§  47) 


(2) 


( 
l- 


122  Orbits  of  Spectroscopic  Binaries  [OH.  xi 

showing  that  rzjr-i  is  of  the  order  of  e.     Hence,  by  division, 

5e2      e4 


r,"   sinOT+  V        24  "96 

rg    cos 


iT  +  &)        /    _  Te3  _  e^  _       \ 
T+A)  =     \        24      96      "V 
and,  by  subtraction  and  addition, 

r,    sin  Q77  +&,-&)      e3       e° 


sn 
sin 


the  last  equation  containing  no  term  in  e5.  Eccentricities  as  high  as  075 
are  met  with  occasionally,  but  even  so  it  is  evident  that  (pT  +  /&  —  ft)  is  a 
very  small  angle  which  can  scarcely  exceed  2°  and  is  generally  negligible. 
If  then 


it  is  possible  to  neglect  a2  and  the  last  equations  become 


T-i  .  (1  +  a  cot  (4/3,  -  2&)}  -  e  -  j 


whence 


From  this  equation  e  is  easily  found  by  trial  and  error,  and  then  a,  which 
gives  T,  is  found  from  (3).  The  equations  (2)  give  Kl  and  K2,  whence  finally 
K  and  <o  are  derived  by  (1).  The  process  is  therefore  very  simple,  even 
without  special  tables,  when  once  the  harmonic  representation  of  the  velocity 
curve  by  two  periodic  terms  has  been  obtained.  This  can  be  done  very 
easily  and  with  all  needful  accuracy  by  taking  a  sufficient  number  of  equi- 
distant ordinates  from  the  curve. 

115.  It  is,  however,  more  usual  in  practice  to  find  approximate  pre- 
liminary elements  by  methods  which  are  largely  graphical  and  to  improve 
them,  if  thought  necessary,  by  a  least-squares  solution  giving  differential 
corrections.  Thus  ZK  is  the  apparent  range  of  the  velocity  curve,  and  when 
the  periastron  point  P1  has  been  located  on  the  curve,  T  is  known,  while  the 
areal  property  which  fixes  the  position  of  the  line  y  =  7  has  been  explained 
(§  113).  The  remaining  elements  to  be  determined  are  therefore  e  and  <o, 
and  these  are  connected  by  the  relation  Ke  cos  &>  =  7'  —  7.  A  number  of 
interesting  properties  have  been  used  for  the  purpose. 

Among  these  are  the  properties  connected  with  a  focal  chord  of  the 
orbit.  Let  ^  be  the  time  at  a  certain  point  of  the  orbit  and  w  and  El  the 


iu-116]  Orbits  of  Spectroscopic  Binaries  123 

corresponding  true  and  eccentric  anomalies.  Let  t2  be  the  time  at  the  other 
end  of  the  focal  chord  through  the  point  and  180°  +  w  and  E2  the  true  and 
eccentric  anomalies.  Then 

(1  -  efi  tan  £«;  =  (!+  e)4  tan  \Elt     ^  (t,  -T)  =  E1-e  sin  El 

—  (1  —  e)   cot  \w  =  (1  +  e)  tan  ^E2,     /j,(t2  —  T)  =  E2  —  e  sin  Ez. 
Hence 

-  (1  _  e)  =  (1  +  e)  tan  ^E1  tan  |#2 

or 

and  therefore 


=  (E2-EJ-  sin  (Es-EJ. 
Also 

tan  £  (#2  -  #0  =  -  J  (1  -  e2)*  e-1  (cot  £w  +  tan  | 

=  —  cot  <f>  cosec  w. 
Hence,  if  2??  =  E.2-Elt 

p  (t2  —  ti)  =  2?;  —  sin  2?7,     tan  <£  sin  w  =  —  cot  77. 

Similarly,  if  £3,  £4  are  the  times  at  the  ends  of  the  perpendicular  chord,  where 
the  true  anomalies  are  90°  +  w,  270°  +  w, 

P  (t\  —  ts)  =  ST/'  —  sin  2?/,     tan  <£  cos  w  =  —  cot  rj '. 

The  angles  17,  77'  are  easily  found,  especially  with  the  help  of  a  suitable  table 
of  the  function  (x  —  sin  #),  and  hence  0  or  e  and  w  =  u  —  w.  But  the  ordinate 
at  the  point  ^  gives  y—<y'  =  K  cos  w  and  therefore  u,  whence  the  value  of  to 
can  be  inferred.  The  equations 

tan  $E!  =  tan  (45°  - 1<£)  tan  £w,  ^(t1-T)  =  E1-e sin ^ 

tan  £  #3  =  tan  (45°  -  £  0)  tan  ( Jw  +  45°),     p(t3-T)  =  E3- e  sin  #:i 
will  give  two  independent  values  of  T. 

Sets  of  four  points  related  in  this  way  are  easily  located  on  the  velocity 
curve,  for  they  are  given  by  y  —  7'=  ±7jTcosw,  ±Ksinu.  Thus  the  four 
points  y  — 7/=±JK"/v2  are  very  suitable  for  the  purpose.  Here  u  =  45°, 
w  =  45°  —  <w.  Two  special  sets  have  been  mentioned  in  §  113,  namely,  AB, 
EF  where  u  =  0°,  w  =  —  co,  and  PjPg,  L^LZ  where  w  =  0°.  In  the  latter  case 
y—y'  =  ±K  cos  a,  ±  K  sin  w,  giving  w  immediately,  ti  =  T,  and  e  is  given 
by  <£  =  77'  -  90°. 

116.  There  are  also  properties  connected  with  a  diameter  of  the  orbit. 
If  E  is  the  eccentric  anomaly  at  a  point,  E+^TT  and  E  +  f  TT  are  the  eccentric 
anomalies  at  the  ends  of  the  diameter  conjugate  to  that  which  passes  through 
the  point.  Let  tl}  tz  be  the  corresponding  tim^s.  Then 


124  Orbits  of  Spectroscopic  Binaries  [CH.  xi 

so  that 

$p  (t,  +  t2  -  2T)  =  E  +  TT 

I/A  (t2  —  t1  —  |P)  =  e  cos  E. 

Now  the  points  G,  D,  in  which  the  line  y  =  7  cuts  the  velocity  curve,  satisfy 
this  condition  and  the  conjugate  diameter  being  parallel  to  the  line  of  nodes 
makes  the  angle  —  o>  with  the  major  axis.  Hence  in  this  case 

—  tan  w  =  cos  $  tan  E 
and  therefore 

$/*  (t,  -t1-$P)  =  e(l+  tan2  &>  sec2  <£)  "  - 

=  e  cos  <y  (1  —  e2  cos2  &>)  ~  -  cos  <f> 
which  gives  e  =  sin  <£  when  e  cos  &>  =  (7'  —  j)/  K  is  known.     Also 

-  e  =  £/&  (*,-«,-  £P)  sec  I/A  («,  +  ^  -  2T7) 
which  gives  a  relation  between  e  and  T. 

Another  pair  of  such  points  is  Klt  K2,  corresponding  to  the  ends  of  the 
minor  axis.     Since  E  =  0  in  this  case, 


Let  Uj,  u2  be  the  longitudes  at  Klt  K2.  Then  the  radial  velocities  at  these 
points,  relative  to  G,  are 

+  %K  (cos  Wj  —  cos  w2)  =  +  K  sin  |  (w?  —  HI)  sin  ^  (z^  +  u1)=±  K  cos  0  sin  o>. 

This  quantity  is  therefore  given  by  the  ordinates  at  K1}  K2  on  the  velocity 
curve,  relative  to  the  line  y  =  7. 

117.  The  velocity  curve  also  possesses  interesting  integral  and  differential 
properties  which  may  be  useful.  It  is  necessary  to  have  a  consistent  system 
of  units,  and  since  those  of  time  and  velocity  have  already  been  adopted,  the 
unit  of  length  is  fixed  and  the  natural  system  is: 

Unit  of  time  =  1  mean  solar  day  =  86400  mean  sees., 
Unit  of  length  =  86400  km.  =  0'0005779  astronomical  units, 
Unit  of  velocity  =  1  km.  per  second, 
Unit  of  mass  =  that  of  the  Sun. 
Now  the  constant  of  areal  velocity  in  the  orbit  is 

p  V2  =  lirabjP  —  pa2  cos  <£ 
so  that 

a  sin  i  =  V2/j.~1  cos  0  sin  i  =  K/jr1  cos  <f>. 

The  argument  relative  to  the  areas  -of  the  velocity  curve  in  §  113  can  now  be 
made  more  precise.  For  the*  tangents  to  the  orbit  at  C  and  D,  referred  to 
the  principal  axes  of  the  ellipse,  are 

x  sin  a)  +  y  cos  a>  =  ±  V(«2  sin2  &>  +  62  cos2  to) 


116-iis]  Orbits  of  Spectroscopic  Binaries  125 

and  the  perpendiculars  on  them  from  the  focus  G  are 

z\  ,  22=  ±ae  sin  &>  +  a  \/(l  —  e~  cos2  &>). 

Measured  from  the  line  y  =  y  let  AI  be  the  area  of  the  velocity  curve  from  A 
to  C,  —  Al  from  C  to  B,  —  A.2  from  B  to  D,  and  ^3  from  D  to  A     Then 


cos  <  -  e  cos  &) 

cos  <f>  .  e  sin  &> 
A  ^2  =  K*pr-  cos4  <£. 

When  Alt  A2  have  been  measured  in  the  proper  units  these  equations  deter- 
mine (j>  (or  e)  and  &>. 

118.     If  the  tangent  to  the  velocity  curve  makes  an  angle  -\/r  with  the 

axis  of  time, 

dV          ,,   .       dw 
tan  \/r  =  -TT  «  —  JL  sin  t*  -rr 

cfa  at 

and  r  being  the  radius  vector  in  the  orbit,  the  constant  areal  velocity  is 

pa?  cos  <f)  =  r2  -f-  . 

Hence 

tan  \IT=—  fjiK  cos  $  sin  u  (a/r)2 

=  —  /j,K  sec3  <f>  sin  u  (1  +  e  cos  w)3 
and  at  special  points  on  the  curve  tan  i/r  has  these  values  : 

A,B     :  ti  =  0°,  180°         :  tan  i|r  =  0 

^,  F     :  it  =90°,  270°      :  tan<f  =  +  ^  sec3  <£(!  ±esino>)2 

Pl}  P2  :w  =  0°,  180°        :  tan  ^  =  +  pK  sec3  <^>  sin  w  (1  ±  e)2 

Xj  ,  Z.,    :  w  =  90°,  270°       :  tan  ty  =  +  //,/f  sec3  ^>  cos  &> 

Kl}  K.2  :  w  =  +  (90°  +  ^>)  :  tan  i/r  =  +  //.X"  cos  ^>  cos  (&>  +  ^>). 


If  tan  i/r  is  found  graphically  at  any  of  these  points,  attention  must  be  paid 
to  the  scales  in  which  ordinates  and  abscissae  are  represented.  These 
expressions  can  then  be  used  in  order  to  find  «o  and  </>. 

Since 

r  oc  (sin  u  cot  ty]  r,     w  =  w  —  &> 


and  u  at  any  point  on  the  velocity  curve  is  given  by  the  ordinate  measured 
from  the  axis  y  =  j',  it  is  possible  theoretically  to  plot  the  actual  orbit  to  an 
arbitrary  scale,  point  by  point.  This  is  scarcely  a  practical  method,  but 
deserves  mention  as  the  counterpart  of  Sir  John  Herschel's  method  for 
double  star  orbits  (§  105). 


126  Orbits  of  Spectroscopic  Binaries  [CH.  xi 

119.  The  values  of  the  elements  found  by  any  of  these  graphical  methods 
are  approximate  only.     They  can  be  improved  by  the  addition  of  differential 
corrections,  SX  to  K,  Be  to  e,  &a>  to  <»,  ST  to  T  ,and  S/j,  to  /A.     Thus  each 
observation  gives  an  equation  of  condition  of  the  form 

V0-  Vc=8y  +  cosu.SK 
and  it  is  easily  found  that 

dw 

-^-  =  sin  w(2  +  e  cos  w)  sec2  <p 

de 

dw 

;c7f,=  —  it  (1  -f  e  cos  wr  sec-*  <p 

dl 

^ 

™  =  (t-T)(l+e  cos  w)2  sec3  <j>. 

It  is  more  usual  to  give  7,  the  radial  velocity  of  the  system,  than  7',  but  this 
quantity  can  be  derived  finally  from  the  relation  7  =  7'—  Ke  cos  &>. 

120.  When  the  elements  of  an  orbit  specified  above  have  been  obtained, 
by  whatever  method,  some  information  can  be  gained  as  to  the  dimensions 
and  mass  of  the  system.     An  equation  already  found  in  §  117  gives 

a  sin  i  =  Kfir1  cos  <£  .  86400  km. 

when  the  unit  of  length  there  adopted  is  explicitly  introduced.  Let  m  be 
the  mass  of  the  star  whose  spectrum  is  observed,  and  m'  the  mass  of  the 
other  star.  Then 


u?a?  (l  +  — ,Y  =  (m  +  m)  C 
\        m  / 


where  C  is  a  constant  depending  on  the  units  employed.     These  being  as 
stated  in  §  117,  the  special  case  when  m'  =  1,  m  =  0,  gives 

47T2  1 

=  (0-0005779)*'     loSC 


It  follows  that 

m'3  (m  +  m')-2  sin3  i  =  [3'81443  -  10]  K3fjL~l  cos3  <j> 

=  [3-01625  -  10]  K3P  cos3  0 

and  it  is  only  this  function  of  the  masses,  involving  the  unknown  inclination 
of  the  orbit,  which  can  be  determined  when  only  one  spectrum  can  be 
observed. 

If,  however,  the  radial  velocity  V  of  the  second  component  of  the  system 
can  be  measured  at  the  same  time,  which  is  possible  when  the  two  superposed 
spectra  are  of  comparable  intensity, 


ii9-i2i]  Orbits  of  Spectroscopic  Binaries  127 

One  such  equation  will  give  the  ratio  m  :  m  when  7  is  known  and  two  will 
give  7  in  addition  without  any  knowledge  of  the  orbit.  It  has  been  supposed 
that  the  radial  velocities  have  been  determined  by  referring  the  stellar 
spectrum  to  a  comparison  spectrum  from  a  terrestrial  source,  as  mentioned  in 
§  111.  When  there  is  no  comparison  spectrum,  as  when  an  objective  prism 
is  used,  and  the  stellar  spectrum  shows  double  lines,  it  is  still  possible  to 
deduce  the  orbit  of  the  system  from  the  relative  displacements  of  corre- 
sponding lines.  But  the  orbit  is  then  the  relative  orbit,  a  is  the  mean 
distance  of  the  components  from  one  another,  and  it  is  easily  seen  that 
(m  +  m'}  sin3  i  must  be  substituted  for  the  above  function  of  the  masses. 

121.  The  true  spectroscopic  binary  cannot  be  resolved  in  the  telescope. 
But  one  or  both  components  of  a  visual  double  can,  when  bright  enough,  be 
observed  with  the  spectrograph,  and  very  interesting  results  can  be  gained 
in  this  way.  Let  a,  a'  be  the  mean  distances  of  the  components  relative  to 
the  centre  of  mass,  expressed  in  terms  of  the  linear  unit  86400  km.  The 
astronomical  unit  contains  1730  such  units.  Let  a"  be  the  visual  mean 
distance  and  -a"  the  parallax  of  the  system,  both  expressed  in  seconds  of  arc. 
Then 

,  ,        mm     ,          ,.  > 

ma  =  ma  =  -  ,  (a  +  a  ) 
m  +  m 


rs      m  +  m 
and  therefore 

V  =  7  -i-  K  (cos  u  -f  e  cos  (a) 

—  7  -f  p<a  sin  i  sec  <£  (cos  u  +  e  cos  o>) 

a" 

=  7-1-  1730  a  sm  i  sec  <f>  (cos  u  +  e  cos  <u)  .  —  -„  .  -      —  ; 

m      m  +  m 

while  for  the  other  component  similarly 

/> 

V  =  7  —  1730  u,  sin  i  sec  d>  (cos  u  +  e  cos  <a)  .  —  ,  .  -  -.  . 

tsr      m  +  m 


a          m' 


If  then  the  elements  of  the  visual  orbit  have  been  independently  determined 
and  the  radial  velocity  of  the  first  component  alone  can  be  observed  at 
different  dates,  the  two  quantities  7  and  (1  +  mjm')  -nr"  can  be  inferred.  If 
the  radial  velocity  of  the  second  component  can  also  be  observed,  the  parallax, 
the  ratio  of  the  masses  and  hence  the  individual  masses  themselves  in  terms 
of  the  Sun  (§  104)  can  also  be  deduced.  From  the  relative  radial  velocity 
alone, 

V  —  V  =  1730  /*  sin  i  sec  </>  (cos  u  +  e  cos  o>)  a" ITS" 

the  parallax  can  be  found,  and  hence  the  total  mass  of  the  system. 

One  question  remains  in  the  determination  of  the  true  orientation  of  a 
double  star   orbit   in  space,  which  can  only  be  decided   by  radial  velocity 


128  Orbits  of  Spectroscopic  Binaries  [en.  xi 

observations.  For  the  spectroscopic  binary  i  has  been  defined  so  that 
0  <  i  <  ^TT,  while  for  the  visual  double  0<t<7r.  This  difference  does  not 
affect  sini,  which  is  positive  in  either  case.  Hence  if  Vlt  V2  are  the 
radial  velocities  of  the  principal  star  at  different  times,  the  two  expressions 

Vl  —  V<2,     cos  (w^  +  to)  —  cos  (w2  +  o>) 

have  the  same  sign,  where  «  is  the  longitude  of  periastron  of  this  star, 
reckoned  from  its  receding  node  in  the  direction  of  motion.  But  X  is  the 
longitude  of  periastron  of  the  companion  at  its  first  node  O  (<  TT).  Hence  if 
the  expressions 

Vl—  V2,     cos  (wx  +  X)  —  cos  (w2  +  X) 

have  the  same  sign,  X  =  &>.  This  means  that  the  principal  star  is  receding 
and  the  companion  is  approaching  when  the  latter  is  at  its  node  fi.  If  on 
the  other  hand  the  expressions  are  of  opposite  signs,  X  =  to  +  TT  and  the 
companion  is  receding  at  fl. 

Otherwise  it.  may  be  possible  to  determine  the  velocities  V,  V  of  the 
principal  star  and  the  companion  respectively  at  the  same  time.     Then  the 

expressions 

V  —  V'f    cos  (w  +  o>)  +  e  cos  o> 

have  the  same  sign,  and  therefore  if  the  expressions 
V  —  V,     cos  (w  +  X)  +  e  cos  X 

have  the  same  sign,  X  =  to,  while  if  they  have  opposite  signs,  X  =  to  +  IT.  The 
same  consequences  follow  as  before.  Thus  a  knowledge  of  either  V1  —  V2  or 
V  —  V  removes  the  ambiguity  with  regard  to  the  true  position  of  the  orbital 
plane,  which  remains  after  the  elements  of  a  double  star  have  been  deter- 
mined from  visual  observations  alone. 


CHAPTER    XII 


DYNAMICAL     PRINCIPLES 

122.  It  will  be  convenient  in  this  chapter  to  recall  some  of  the  salient 
features  of  dynamical  theory  and  to  consider  as  briefly  as  possible  the  form 
of  those  transformations  which  are  of  the  greatest  importance  in  astronomical 
applications.  We  shall  start  from  Lagrange's  equations. 

Let  the  system  consist  of  a  number  of  particles  whose  coordinates  can  be 
expressed  in  terms  of  n  quantities  ql}  q2,...,qn  and  possibly  of  the  time  t. 
Let  m  be  the  mass  of  a  typical  particle  situated  at  the  point  (x,  y,  z). 

Then 

doc      dx  dx 

x  =  3-  +  5—  .  QI  +  .  .  .  +  5  —  .  qn 

dt      dq,  dqn 

so  that 

dx  __  doc 

dqr      dqr  ' 
Hence 

d  (.      dd?\          d  f.  dx\ 


d  (.  dd?\  d  f.  dx\ 
-j-  *m  =-r-  )  =  m  -j-  (x  ~—  1 
dt  \  dqJ  dt  \  dqrJ 


=     ~  --  -     a*— 
dqr  dqr 

where  X  is  the  component  of  the  force  acting  on  m.     If  T  is  the  kinetic 
energy  of*  the  whole  system, 


Hence  adding  all  the  equations  of  the  preceding  type  for  the  three  co- 
ordinates and  all  the  particles, 


~r.    ^rr-     —         -      ^  --  h  J.    «  —  T  ^  ^  —     +  ~  —  . 
dt  \oqrj          \      dqr          dqr          dqr/      dqr 

Now  the  forces  which  occur  in  astronomical  problems  are  in  general  con- 
servative, and  we  can  write 

2  (Xdx  +  Ydy  +  Zdz)  =  -dU 

P.  D.  A.  9 


130  Dynamical  Principles  [CH.  xn 

where  dll  is  a  perfect  differential.  U  represents  the  work  done  by  the 
forces  in  a  change  from  the  actual  configuration  to  some  standard  configu- 
ration and  is  called  the  potential  energy.  We  therefore  have  . 


d_  /dT\  =  d(T-U) 
dt    d)  d 


dt  \dqrJ  dqr 

But  U  does  not  contain  qr,  and  hence,  if  we  write  T  =  U  +  L,  this  becomes 
^  (dL\=  dL  9 

which  is  the  standard  form  of  Lagrange's  equations. 

The  function  L  is  often  called  the  Kinetic  Potential.     In  the  absence  of 

moving  constraints  (or  some  analogous  feature)  within  the  system  —  =  ...  =  0. 

ot 

Then  T  is  a  homogeneous  (positive  definite)  quadratic  form  in  q1} ...,  qn. 

123.     If  L  does  not  contain  £  explicitly,  the  equations  admit  an  integral 
called  the  Integral  of  Energy.     For  in  this  case 

dL=2(<M     .      dL_ 
dt       r  \dqr'          dqr' 
=      '^/3X\    .   +dL_ 


dL 

^~  (    ^ : 

dt  \  r  dqr 
so  that 


(2) 


where  h  is  a  constant  of  integration.     Replacing  L  by  T—  U,  where  T  is  a 
homogeneous  quadratic  form  in  qr  and  U  does  not  contain  qr,  we  have 

h  =  2T-(T-  U)  =  T  +  U 
which  shows  that  h  is  the  sum  of  the  kinetic  and  potential  energies. 

More  generally,  let  L  contain  t  explicitly  through  U  and  let  T  no  longer 
be  a  homogeneous  function  in  qr  but  of  the  form  T2  +  Tl  +  T0,  where  T2  is  a 
homogeneous  quadratic  function,  Tl  a  linear  function  and  T0  of  no  dimensions 
in  qr.  Then  similarly 

dL      d  I ' ^  dL   .  \      dL 
dt      dt 

d 


dt  dt 

or  since  L  =  T.2  +  Tl  +  T0  —  U 

&     /  ni  rn       .      TT^  OU 


122-124]  Dynamical  Principles  131 

an  equation  which  applies  to  relative  motion.     When  U  does  not  contain  t    • 

T2-T0+U=h. 
When  U  does  contain  t  the  equation 

J  dt 

is  a  purely  formal  integral  because  it  is  to  be  understood  that  any  coordinates 
occurring  in  dU/dt  are  expressed  in  terms  of  t  before  integration.  This 
implies  a  knowledge  of  the  complete  solution  of  the  problem.  But  the 
equation  is  not  without  its  uses.  Thus  if  U=  U0  -f  U',  where  U0  does  not 
contain  t  and  the  effect  of  U'  is  small  in  comparison  with  the  effect  of  U0, 
preliminary  values  of  the  coordinates  in  terms  of  t  may  be  found.  When 
these  are  inserted  in  dU' /dt  a  closer  approximation  to  the  true  integral  will 
be  obtained  and  the  process  can  be  repeated.  The  true  meaning  of  the 
equation  is  therefore  connected  with  a  method  of  approximation. 

124.  The  above  form  (2)  of  the  integral  of  energy  is  directly  connected 
with  the  Hamiltonian  form  of  the  equations  of  motion  whereby  the  n 
Lagrangian  equations  of  the  second  order  are  replaced  by  a  system  of 

2n  equations  of  the  first  order.     For  we  may  write 

t 

ar  ar 

v  •  r      IT 

The  n  equations  for  pr  are  linear  in  qr  and  when  solved  express  qr  in 
terms  of  (qr,  pr),  this  symbol  being  used,  where  no  ambiguity  is  to  be  feared, 
to  denote  all  the  quantities  qlt  q2,....,  qn>  PD  PZ>--->  Pn-  Hence  L  and  H  can 
be  expressed  either  in  terms  of  (qr,  qr)  or  of  (qr,  pr).  Thus 

^  T      ^  dL    5,         ^  dL     ~ . 
oL  =  2t  ~ —  .  oqr  +  2t  ^-7-  .  oqr 
r  oqr  r  eqr 

re .  3l£     «,.  *dL 

o2^qr  ?— -  =  2i  qr  opr  -f-  2i<  ^rr-  •  odr 

dqr      r  r  dqr 

and  therefore 

BH  =  2  (qr  8pr  —  pr  Bqr) 

r 

since 

dt  \dqrj      dqr ' 
It  follows  that 

»  a  IT  3IT 

'    —    —         '    =—  —     f    =1    2  ^  (1} 

dpr '  dqr  ' 

and  this  is  the  form  of  the  equations  which  is  called  canonical. 

When  L  has  its  natural  form,  H=  T+  U.  If  L  does  not  contain  t  ex- 
plicitly, neither  does  H,  and  the  integral  of  energy  (2)  becomes  simply  H=  k. 

9—2 


132  Dynamical  Principles  [CH.  xn 

125.     Let  us  consider  the  differential  form 

dO  =  2  pr  dqr  —  Hdt 

r 

or 

d(^prqr  —  0)  =  ^  qr  dpr  +  H  dt. 

r  r 

If  dd  is  a  perfect  differential,  the  right-hand  side  of  both  equations  must 
also  be  perfect  differentials,  and  this  -requires  that 

dpr  _      dH      dqr  _  dH 
dt  dqr'      dt       dpr 

or  the  canonical  equations  must  be  satisfied.     Let  us  suppose  now  a  trans- 
formation from  the  variables  (qr,  pr)  to  the  variables  (Qr,  Pr)  such  that 

.  .......  (4) 


where  dW  is  a  perfect  differential  and  W  is  expressible  either  in  terms  of 
(qr,  Pr)  or  of  (Qr,  Pr).  Such  a  transformation  is  called  a  contact  transforma- 
tion, or  in  the  particular  case  when  (qr)  can  be  expressed  in  terms  of  (Qr) 
alone  [by  relations  not  involving  (pr)  or  (Pr)]  an  extended  point  transformation. 
If  W  contains  t  in  addition  we  may  write 

dW  dW 

2  Pr  dQr  —  ^pr  dqr  —  3—  .  dt  =  —  d  W  —  -~—  .  dt 

j*  T»  Ov  Ov 

so  that  when  dd  is  introduced 


Each_  side  of  this  equation  is  a  perfect  differential  provided  d6  is  a  perfect 
differential,  and  in  this  case 


where 


dK       *  _  dK 

~dQr'       ^~dPr' 


Since  these  equations  equally  with  the  form  (3)  express  the  conditions 
required  if  dd  is  to  be  a  perfect  differential,  they  must  be  equivalent  to  (3). 
Thus  we  see  that  any  transformation  of  variables  satisfying  the  condition  (4) 
leaves  the  equations  of  motion  in  the  canonical  form. 

126.     In  consequence  of  (4) 

dW  _dW 

~Wr'       Pr~3qr' 

Hence  K  will  vanish  in  virtue  of  (6)  provided 

dW 
+        =  0  ..................  (8) 


125-127]  Dynamical  Principles  133 

This  equation  is  known  as  the  Hamilton- Jacobi  equation.     But  when  K  =  0, 

-Pr  =  &,       &  =  «*• 

where  ar  and  (3r,  by  (5),  are  arbitrary  constants.  Hence  if  any  function  W 
can  be  found  which  satisfies  (8)  and  contains  n  arbitrary  constants  (ar)  in 
addition  to  (qr)  and  £,0the  solution  of  the  problem  is  completely  expressed  by 
the  2n  equations  (7)  written  in  the  form 

SW  3W 

a^'-A.  ft-aj- <»> 

where  (/3r)  are  n  additional  arbitrary  constants. 
If  H  does  not  contain  t  explicitly  we  may  write 

W=-ctnt+W 

where  W  is  a  solution,  containing  (n  —  1)  constants  (ar)  apart  from  an  but 
not  t,  of  the  equation 


The  solution  (9)  is  therefore  replaced  by 

(H) 


127.     In  the  set  of  equations  (7)  W  is  an  arbitrary  function  of  (Qr,  qr). 
Instead  of  making  W  a  solution  of  (8)  let  it  satisfy  the  equation 


where  H0  is  the  Hamiltoniau  function  of  another  problem  also  presenting 
n  degrees  of  freedom.     Hence  as  before 

P-r  ~  @r,       Qr  =  <*r 

where  (cer,  {3r)  are  the  2n  arbitrary  constants  of  the  problem  defined  by  H0. 
Hence  the  equations  (5)  and  (6)  become 


where 

3F 

dt  0> 

Thus  if  the  H0  problem  has  been  solved  and  the  constants  of  a  solution  of 
the  corresponding  Hamilton-Jacobi  equation  are  known,  the  same  form  of 
solution  applies  to  the  H  problem  with  the  difference  that  the  quantities 
which  remain  constant  in  the  first  problem  undergo  variations  in  the  second 


134  Dynamical  Principles  [OH.  xn 

problem  which  are  defined  by  (12).  This  is  the  foundation  of  Lagrange's 
method  of  the  variation  of  arbitrary  constants.  The  simple  form  of  (12) 
depends  essentially  on  the  function  K  being  expressed  in  terms  of  the 
constants  which  occur  in  a  solution  of  a  Hamilton-Jacobi  equation  and 
which  may  be  called  a  set  of  canonical  constants. 

If  we  suppose  that  the  problem  defined  by  H0  has  been  solved  by  some 
other  method  than  through  the  medium  of  a  Hamilton-Jacobi  equation,  a 
different  set  of  constants  will  be  obtained.  Let  Am  be  a  typical  member  of 
such  a  set.  Then  Am  is  some  function  of  (ar,  f3r).  Hence 


dA 


-^        •  o/~>  "i  ,-.     '  ~ 

dar    d/3r       d/3r    darj 
s 


—  —  —  -          —  — 
dar  'dAg'd/3r      3/3r  'dAs'da.r 

-  1  (  A       A  }  ^K 

-  Z  [Am,  A.J  ^ 

where  K  =  H  —  H0  as  before,  and 


a  form  of  expression  which  will  be  defined  later  (§  130)  as  a  Poisson's  bracket. 
128.     Let  us  consider  the  integral 

J=ft>  Ldt=  (  (T-  U)dt 

Jto  J  t0 

=  !f>(-H+2prqr)dt    ........................  (13) 

J  t0 

by  the  first  set  of  equations  in  §  124.     We  have  therefore 

-  SH  +  2rS 


where  S  denotes  a  change  in  (qr,  pr)  but  leaves  t  at  each  point  unaltered. 
Hence  8J  =  0  if  8qr  =  0  at  the  limits  and  if  the  canonical  equations  are 
satisfied.  And  this  proves  Hamilton's  principle  that  in  the  passage  from 
one  fixed  configuration  to  another  the  integral  J  has  a  stationary  value  for 
the  actual  motion  as  compared  with  any  other  neighbouring  motion  in  which 
the  time  at  corresponding  points  is  the  same. 


127-129]  Dynamical  Principles  135 

If  however  8  denotes  a  change  in  t, 

8J  =  _  8  t  '  Hdt  +  8  I  %prdqr 

J  t0  .'0 


o 

Hence  when  two  neighbouring  forms  of  motion,  each  compatible  with  the 
canonical  equations,  are  compared,  the  complete  variation  between  two 
positions  0  and  1  is 

8J=  IZprSq^      \H8t\\ 

Accordingly,  if  the  initial  time  is  taken  as  fixed  and  (<xr,  j3r)  are  the  initial 
values  of  (qr,  pr),  we  have 

dJ  _  dJ__ 

dqr~Pr>      far~      * 
and 


But  this  is  the  Hamilton-  Jacobi  equation.  Hence  the  integral  J"  is  a  par- 
ticular solution  of  this  equation.  And  further,  since  we  have  reproduced  the 
equations  (8)  and  (9)  of  §  126  except  that  J  is  written  in  the  place  of  W,  we 
see  that  J  is  that  solution  which  contains  the  initial  values  of  the  coordinates 
as  its  n  arbitrary  constants. 

129.     Let  us  suppose  now  that  H  does  not  contain  t  explicitly,  so  that 
the  integral  of  energy  H  =  h  exists.     Then  if 


=  r 

J  t0 


........................  (14) 

t0 

i1    r*1' 

r     +       (Zqr8pr  -  2,pr8qr)  dt. 

JO       Jtv 


But 

2qr8pr  -  2pr8qr  =  2  JT-  8pr  +  2  ^—  8qr 
opr  oqr 

=  8h 
and  therefore 


Sh .  dt.     . 

This  is  the  complete  variation  of  J  and  it  vanishes  between  fixed  terminal 
points  if  8h  =  0  in  each  intermediate  position,  i.e.  if  the  time  is  assigned  to 
each  displaced  position  in  such  a  way  that  the  equation  H  =  h  is  satisfied  in 
the  varied  motion.  Under  these  conditions  the  integral 


f'(T-  U+h)dt 
J  t 


136  Dynamical  Principles  [OH.  xn 

has  a  stationary  value  in  the  course  of  the  actual  motion  as  compared  with 
motion  in  any  neighbouring  paths. 

This  integral  is  called  the  action  and  the  proposition  established  is  known 
as  the  principle  of  least  action.  When  T  is  a  quadratic  function  of  the 
velocities  h  —  T+U  and  the  integral  becomes 


..............................  (15) 

t0 


and  in  problems  which  involve  only  one  material  particle  this  is  simply 

rt,  ri 

/==      v2dt=\   vds  ...........................  (16) 

J  to  JO 

where  v  is  the  velocity  of  the  particle  (of  unit  mass). 

The  integrals  which  we  have  found  to  be  stationary  are  not  necessarily 
minima.  The  necessary  conditions  in  order  that  an  integral 

rti 

J  =         f(lr,  qr)dt 
*° 

shall  be  an  actual  minimum  are  : 

(1)  The  first  variation  8J  vanishes  between  fixed  terminal  points. 

(2)  The  function  of  (er) 

rif 
0  M  =f(qr,  qr  +  er)-^€r^~ 

is  a  minimum. 

(3)  Between  the  terminal  positions  0  and  1  no  intermediate  position  P 
exists  such  that  0  and  P  can  be  joined  by  a  neighbouring  path  which  satisfies 
the  dynamical  conditions  and  is  other  than  the  path  considered.    The  nearest 
point  to  0  on  the  path  which  does  not  satisfy  this  condition  is  called  the 
kinetic  focus  of  the  point  0. 

130.  It  is  necessary  to  study  the  properties  of  certain  expressions 
connected  with  the  transformations  which  are  frequently  employed.  Let 
ul}  HZ,  ...  ,  u2n  be  2n  distinct  functions  of  (qr,  pr).  The  first  expression  is 


dum'  du 

which  is  called  a  Lagranges  bracket  and  is  denoted  by  [HI,  MM].     The  second 
expression  is 

dum     dum   duA      ^  8  (MJ,  um) 

.  •=  --  -r  -  .-  —  \  = 


.  .  ,. 

dqr    dpr       dqr    dprj      r   0  (qr,  pr) 

This  is  called  a  Poissons  bracket  and  will  be  denoted  here  by  the  symbol 
[ui>  um}.     It  is  evident  that  we  have 

[MI,  um]  =  -  [um,  ui],      (1  4=  m)      . 

{ui,  um]  =  -{um,  ui},      (I  4=  m) 


129,   130] 


Dynamical  Principles 


137 


There  are  also  relations  between  the  two  types  of  expression,  and  these 
we  shall  now  investigate. 

Let  two  linear  substitutions  be  defined  by 


and 


•  zm , 


where  r  can  have  all  values  l,...,n  and  I  and  m  can  have  all  values  1, ...  , 
The  result  of  eliminating  yr,  yn+r  is  to  give 


•  o 

dumj 


2n 

2  [HI,  Um]  Z1t 


.(19) 


But  the  substitutions  can  be  reversed  by  writing 


zm  = 


TWr 

•"  a^r  •  y»- — 2 


yn+r  • 


The  equivalence  of  these  forms  is  easily  verified  since 

au,  agri  _      "» ra^  a^i  _ 

"5      •  ^\          —  •*•'       *•  I  5      •  'N          —  ">  •  •  •  • 

8<?r  «*ij  ?  L8?*-  9wd 

When  yr,  2/n+r  are  eliminated,  these  give 

^      "  /3wz  9wm     8ww  3 
^tn  =  2  act  2  U-  .  -5 -5 — .  a 


.(20) 


The  resultant  substitutions  (19)  and  (20)  must  therefore  be  equivalent,  and 
accordingly  their  determinants,  written  in  the  forms 


and 


{«*,,     Wi},  {M!,     Ma},...,  {MU 
{^2,     Wa},  (W2,    Ma},...,  {Ma, 


(21) 


are  reciprocal.  This  means  that  any  constituent  of  either  determinant  is 
equal  to  the  co-factor  of  the  corresponding  constituent  in  the  other  determinant 
divided  by  that  determinant.  Any  Lagrange's  bracket  is  thus  expressible  in 
terms  of  Poisson's  brackets,  and  vice  versa. 


138  Dynamical  Principles  [CH.  xn 

131.     Let  us  now  consider  the  explicit  conditions  for  a  contact  trans- 
formation.    We  have  in  this  case 


r  r  r  r    I 

a  perfect  differential.     Hence 


ap« 

always,  and 


unless  I  =  m,  in  which  case 


It  is  at  once  evident  that  these  conditions  may  be  written 

[P,,Pm]  =  0,     [Qf,  QM]  =  0 
for  all  values  of  /  and  w, 

[Qi,  ^m]  =  0 

for  all  unequal  values  of  I  and  m,  and 

[Qi,  PI]  =  I 

for  all  values  of  I.  In  other  words,  in  the  case  of  a  contact  transformation 
all  the  Lagrange's  brackets  vanish  with  the  exception  of  those  which  are  of 
the  form  [Qt,  PI],  and  these  are  all  unity. 

Let  us  now  put 

Ur=Qr,       Un+r  =  Pr,       (r=l,  2,  ...,  W). 

Then  the  substitution  (19)  becomes  simply 

Xr  =  Zn^.r,       ^n-^r =        %r- 

But  this  shows  that  all  the  Poisson's  brackets  occurring  in  (20)  vanish 
except  those  which  are  of  the  form  [HI,  ui*n],  and  these  may  be  written 

{Qt,  Pt\  =  1  or  {Plt  Qt]  =  -  I.' 

The  conditions  for  a  contact  transformation  are  therefore  of  the  same  simple 
form  whether  expressed  in  terms  of  Lagrange's  or  of  Poisson's  brackets. 

Again,  the  substitutions  of  §  130, 


131,  132]  Dynamical  Principles  139 

become  identical  when  m  =  n  +  I,  since  zn+i  =  xi.     Hence 

dqr^djj.      d_Pr=_<tij 

<)Qi    dp/   aQi       fyr' 

But  when  I  =  n  +  m,  they  are  identical  except  for  an  opposite  sign  throughout, 
since  xn+m  =  —  zm,  and  thus 


_ 

W^n~      dpr'     dPm~dqr' 
These  relations  hold  for  all  values  of  I,  m  or  r  not  exceeding  n. 

132.     Let  us  consider  the  transformation 

Qr  =  qr  +  €?/,      Pr  =  Pr  +  & 

where  qr',  pr'  are  any  functions  of  (qr,  pr)  and  e  is  an  infinitesimal  constant. 
If  the  transformation  is  an  infinitesimal  contact  transformation, 
d  W  =  2  {(pr  +  epr')  d  (qr  +  eqr')  -  prdqr} 

r 

=  €  S  (pr'dqr  +  Prdqr) 

r 

is  a  perfect  differential.     Hence  we  may  write 

e  2  (pr'dqr  -  qr'dpr)  =  d  (  W  —  e  ^prqr') 

r  r 

=  -e.dK 
where  K  may  be  any  function  of  (qr,  pr).     Accordingly 

,  _  dK         ,  _     dK 
qr~dp-r'     Pr=    ~dqr 

and  the  general  form  of  an  infinitesimal  contact  transformation  is  given  by 

9-fiT  dK 

«'-«'+«^-   *'-*  —  a£  ..................  (22) 

where  ^T  is  an  arbitrary  function  of  (qr  ,  pr). 

If  for  e  we  write  8t,  the  equations  (22)  become 

Sqr=dl{       8pr=_dK 

Bt       dpr  '      Bt  dqr 

and  comparing  this  form  with  that  of  the  canonical  equations  of  motion  we 
see  that  the  progressive  motion  of  a  system  from  point  to  point  corresponds 
to  a  succession  of  infinitesimal  contact  transformations. 

The  effect  of  substituting  (Qr,  Pr)  in  any  function  f  of  (qr,  pr)  is  to 
produce  an  increment 


(23) 


140  Dynamical  Principles  [CH.  xu 

133.  Let  us  consider  a  disturbed  motion  in  which  (qr,  pr)  become 
(qr  +  Bqr,  pr  +  8pr}  at  the  time  t.  If  this  motion  is  compatible  with  the 
canonical  equations 

dH  dH 

Qr  —  5 —  >       Pr  —  —  o — 

dpr  dqr 

we  must  have 


d  ^    ,      WJ^L    x          d*H     s    \ 
T  (Sqr)  =  2,    5 — 5-  .  bqg  +  5 — =—  .  6ps 

WV  «    Vdpr^s  3/>r9p«  / 


dt 

with  similar  equations  for  Bpr.  Now  let  us  suppose  that  the  new  variables 
are  those  given  by  (22).  These  will  lead  to  a  particular  solution  of  the 
varied  motion  provided 

d 


dt\dprj      s  \dprdqs'dps      dprdps' dqsj 

-A  v  f^[  ?^_^"  ?K\ 

dpr  s  \dqg'dpg      dpg'dqs) 
,-,  /dH    (PK       dH    cPK 

—  2t  I . . 

__a_s  /_  .  dK_  .  dK\ 


+ 


=  !_/^_^    A@x\~lL@K} 

\dt        dt  )      dt  \dpj      dt  \dpj 


- 

dpr  \  dt  J 

with  a  similar  set  of  conditions  arising  from  the  equations  for  Bpr.  But 
it  is  evident  that  all  these  conditions  will  be  satisfied  if  K  is  an  integral 
of  the  system,  for  then  K  =  0.  We  thus  see  that  if  K  is  an  integral,  the 
equations  (22)  are  a  particular  solution  of  the  equations  for  the  disturbed 
motion. 

134.     Let  u  be  another  integral  of  the  undisturbed  system.   Then  u  +  AM 
must  also  have  a  constant  value  in  the  disturbed  motion.     But  by  (23) 

AM  =  e  {u,  K} 

when  the  disturbed  motion  is  that  obtained  by  the  infinitesimal  contact 
transformation  derived  from  K.  Hence  {u,  K}  must  be  constant,  and  we 
have  Poisson's  theorem :  if  u  and  K  are  two  integrals  of  a"  system,  the 
Poisson's  bracket  {u,  K}  is  also  an  integral.  It  might  be  supposed  that  a 
knowledge  of  two  integrals  would  thus  lead  to  the  discovery  of  all  the 


133,  134]  Dynamical  Principles  141 

integrals  of  a  problem.  This  is  not  so  in  general.  The  known  integrals  are 
more  often  of  a  generic  type,  particularly  in  the  case  of  those  gravitational 
problems  with  which  we  have  to  deal,  and  fall  into  closed  groups.  For 
example,  if  we  start  from  two  integrals  of  area  we  obtain  by  Poisson's  theorem 
the  third  integral  of  the  same  type  and  no  further  progress  can  be  made  in 
this  way.  In  order  to  obtain  fresh  information  it  is  necessary  to  start  from 
integrals  which  are  special  to  the  problem  considered. 

Let  u1}  u2, ... ,  u^i  be  2n  distinct  integrals  of  the  problem.  Then  each 
Poisson's  bracket  of  the  type  {ur,  us}  is  constant  throughout  the  motion.  But 
we  have  seen  in  §  130  that  a  Lagrange's  bracket  [ur,  iig]  can  be  expressed  in 
terms  of  all  the  Poisson's  brackets.  Hence  \ur,  Ug]  is  also  constant  through- 
out the  motion.  But  this  gives  no  means  of  finding  additional  integrals  of 
the  problem,  for  in  order  to  calculate  [ur,  us~]  it  is  first  necessary  to  express 
(qr,  Pr)  in  terms  of  the  2n  integrals  (ur).  And  this  presupposes  that  the 
problem  has  been  completely  solved. 


CHAPTER  XIII 


VARIATION    OF   ELEMENTS 

135.  The  Hamilton-Jacobi  equation  corresponding  to  elliptic  motion 
about  a  fixed  centre  of  attraction  is  very  simply  solved  when  the  variables 
are  expressed  in  polar  coordinates  (r,  I,  X),  so  that  (I,  X  having  the  same 
relation  to  one  another  as  longitude  and  latitude) 

qi  =  r,     £2  =  X,     q3  =  I. 

Then,  after  suppressing  the  factor  m  in  the  potential  energy  U  and  therefore 
treating  the  mass  factor  in  the  momenta  as  unity, 

U  =  -  fir-\    //,  =  &2  (1  +  m)  =  ri*a3 


PI    —      ^»    PZ  =  r^->    PS  —  r"  c°s2  X  .  / 

H  =      T  +  U  =  $  (p*  +  r~%*  +  r~2  sec2  X  .  p./)  -  pr~\ 

The  Hamilton-Jacobi  equation  (§  126)  therefore  takes  the  form,  since  H  does 
not  contain  t, 

/8FV      1/8^Y+       L.    £5.Y_2a+2'i 
(dr  )  +  r«Ux  )  +  r2cos2xl  dl  )  ~    "*l  +  r 

where   W=W'  —  a.lt.     Integration  by  separation  of  the  variables  is   then 
easy.     For 

/8FV 


obviously  satisfy  the  equation.     Hence 

Tir/       fr  /«        2/i     a22\     7 
"    =  I    (  2«j  +  --  — -  I   ar  + 


135,136]  Variation  of  Elements  143 

is   an  integral   which   contains   the   three   independent  constants  «1}  a.2,  «3. 
Therefore  the  complete  solution  of  the  problem  is  given  by  the  equations 

2,11        nz\~% 

+-     * 


dW  fx  _i 

—  /33  =  -^  —  =  Z  —      «.,  sec2  X  (a,,2  —  «3'2  sec2  X)    2  c£X 
o«3  /• 

where  /9j,  /32,  /33  are  three  additional  constants.  The  lower  limit  r0  is  also 
arbitrary.  It  may  be  identified  with  the  pericentric  distance,  and  then  the 
integrals  depending.  on  r  will  vanish  at  the  pericentre. 

136.  We  have  now  to  determine  the  meaning  of  the  six  constants  of 
integration.  Since  the  integral  in  the  first  equation  vanishes  at  perihelion, 
&  is  clearly  the  time  at  this  point.  Also,  by  the  same  equation, 

f..?*-^ 

r       r2 
=  2«!  (r  —  rx)  (r  —  r2)/r2. 

But  at  an  apse,  r  =  0  and  r  =  a  (1  +  e).  These  then  are  the  values  of  rl}  r2, 
and  hence 

/i  =  —  2a  «!,     a./  =  —  2a2  (1  —  e-)  otj 
or 

«!  =  -  fjL/2a,      a2  =  \/{/ia  (1  -  e2)}. 

Also  if  we  put  or3/a2  =  cos  i  the  second  and  third  equations  become  on 
integration 

-  /32  =  —  /!  (r)  +  sin"1  (sin  X/sin  i) 

—  /33  =      £  —  sin"1  (tan  X/tan  i) 
or 

sin  X  =  sin  i  sin  {/i  (r)  —  /82} 

tan  X  =  tan  i  sin  (£  +  /33). 

This  last  equation  shows  that  the  motion  takes  place  in  a  fixed  plane  making 
the  angle  i  with  the  plane  X  =  0,  which  may  be  taken  to  represent,  for 
example,  the  ecliptic,  with  I  and  X  as  the  longitude  and  latitude  of  the 
planet.  Thus  the  meaning  of  «3  =  o2  cos  i  is  defined,  and  —  /33  is  simply  the 
longitude  of  the  node.  The  preceding  equation  then  shows  that  /,  (r)  —  yS2  is 
the  angle  between  the  radius  vector  of  the  planet  and  the  line  of  nodes, 
i.e.  the  argument  of  latitude.  But  at  perihelion  the  integral  /i  (r)  vanishes. 
Hence  —  /32  is  simply  the  angle  in  the  orbit  from  the  node  to  perihelion, 
or  ta  —  O  in  the  ordinary  notation.  The  canonical  elements  which  we 


144 


Variation  of  Elements 


[CH.  XIII 


have  introduced  can  therefore  be  expressed  in  terms  of  the  usual  elements 
(T  being  reckoned  from  the  epoch  when  the  mean  longitude  is  e)  thus  : 


e2)]  cos  i,      /33  =  —  H. 

The  homogeneity  of  these  constants  will  be  increased  by  introducing  a  = 
instead  of  a,.     This  makes  2^  =  -^  /a?  and   W=  W  +  /x2i/2a2.     Hence 
will  be  replaced  by  ft,  where 


9a 


a3  V 


Since  the  integral  vanishes  at  perihelion,  and  t  =  T  at  this  point, 

/s-e*:.   /e..r-«r—  «  +  .. 

a3       V  a 

The  other  constants  are  easily  seen  not  to  be  affected  by  the  change  in  alf 
ft,  which  can  accordingly  be  replaced  by 


where  e  is  the  mean  longitude  of  the  planet  at  the  time  t  =  0. 

137.  The  expressions  for  a,  «2,  a3,  ft,  ft,  @3  in  terms  of  the  ordinary 
elliptic  elements  which  have  just  been  found  make  it  very  easy  to  calculate 
the  Lagrange's  brackets 

r       ,     v/8a  d/3     d/3  da\ 
[u,  v]  = 


.  — 
u   dv 


.^ 
dv 


where  u,  v  are  any  pair  of  the  six  elements  a,  e,  i,  O,  tn-,  e.  Since  a,  a2,  a.,  are 
functions  of  a,  e,  i  alone  and  ft,  /32)  /33  are  functions  of  U,  -nr,  e  alone,  the 
Lagrange's  bracket  for  any  pair  of  either  set  of  three  elements  vanishes.  It 
is  equally  evident  on  inspection  that  [e,  e],  [i,  or]  and  [i,  e]  also  vanish,  the 
two  constituents  never  occurring  in  a  corresponding  pair  of  canonical  constants. 
Hence  the  complete  array  of  Lagrange's  brackets  may  be  set  out  thus  : 


a 

e 

* 

11 

•or 

6 

a 

0 

0 

0 

[a,  H] 

[a,  tsr] 

[a,  e] 

e 

0 

0 

0 

[e,  fl] 

h»] 

0 

i 

0 

0 

0 

[».  ft] 

0 

0 

n 

-  [a,  ft] 

-  [e,  n; 

1   -[»,n; 

0 

0 

0 

•nr 

—  [a,  «r] 

-!>,» 

1       o 

0 

0 

0 

e 

-  [«>  e] 

0 

0 

0 

0 

0 

136-138] 


Variation  of  -Elements 


145 


where  the  first  constituent  of  each  bracket  taken  positively  is  placed  in  the 
column  on  the  left  and  the  second  constituent  in  the  line  at  the  top.  The 
brackets  in  the  second  diagonal  really  contain  only  one  term  and  are  at  once 
seen  to  be 

[a,  e]  =  -  £  V/A/O, 

[i,  ft]  =  Vytt«  (1  —  e*) .  sin  i 

while  the  remaining  three  brackets  contain  two  terms  and  are 
[a,  ft  ]  =  i  V(  1  —  e2)  /JL/CL  (1  —  cos  i) 


4[e,  O]  =  -  e  *Sjta  (1  -  cos  t)/Vl  —  e2. 

The  value  of  the  whole  determinant  depends  simply  on  the  constituents  in 
the  second  diagonal  and  is  evidently 

A  =  [a,  e]»  [e,  «r]a  [i,  H]2 


138.  It  is  now  easy  to  form  the  reciprocal  determinant,  the  constituents 
of  which  are  the  Poisson's  brackets  of  pairs  of  elements.  On  account  of  the 
large  number  of  zeros  in  the  above  determinant  a  corresponding  number  of 
minors  vanish  and  the  rest  can  be  calculated  without  difficulty.  It  can  in 
fact  be  verified  by  simple  inspection  that  the  reciprocal  determinant  takes 
the  form : 

a  e  i  ft  tx  e 


a 

0 

0 

0 

0 

0 

[a,  i\ 

e 

0 

0 

0 

0 

{*•] 

{*>  61 

i 

0 

0 

0 

{i,  ft} 

{i,  r*} 

(»>  el 

ft 

0 

0 

-  {i,  0} 

0 

0 

0 

-or 

0 

-{«•• 

r}    -  {i,  v] 

0 

0 

0 

e 

-  [a,  € 

1  -  {«,  *} 

-M 

0 

0 

0 

the  first  constituent  of  each  bracket  (written  positively)  being  indicated  in 
the  column  on  the  left  and  the  second '  constituent  in  the  top  line  as  before. 
It  is  also  clear  that  the  partial  substitutions  (§  130) 

#!  =  [a,  li]  z4  +  [a,  tar]  z5  +  [a,  e]  z6 
#2  =  [e,  fi]  £4  +  [e,  -sf]  z5 


=  [i,  O]  z4 


P.  D.  A. 


10 


146  Variation  of  Elements  [OH.  xin 

and 

£4  =  I*.   ft}  a':- 

Z5  =  {e,  -or}  #2  +  {*',  w}  xs 

z6  =  {a,  e}  #1  +  {e,  e}   #2  +  ft,  ej  #3 

must  be  equivalent,  and  it  readily  follows  that 
'   {a,  e}  =       !/[«,*]  =  -2  V5/£ 
e  <n-  =     !       «    =  Vl  -  e«e  V*a 


{i,  ft}  =      I/O',  ft]  =  l/\Va(l-e2)  sin  i 
{e,  e]   =  -  [a,  w]/[a,  e]  [e,  tsr] 


{t,«}=  -[«,«]/[«,«][»,  ft] 

=      (1  -  cos  i)/VfM  (1  —  e2)  sin  i 
{t,  e}    ,     -  {[a,  ft]  [e,  w]  -  [e,  ft]  [a,  «]}/[«,  e]  [e,  «r]  [i,  ft] 

=     (1  —  cos  »)/V/*o  (1  —  e2)  sin  i. 
The  six  Poisson's  brackets  are  thus  all  known. 

139.  A  solution  of  the  Hamilton-Jacobi  equation,  involving  the  six 
arbitrary  constants  a,  a2,  «3,  /3,  ^S2,  j3s,  has  been  found  for  the  case  of  un- 
disturbed elliptic  motion  relative  to  the  Sun.  When  the  action  of  the  other 
planets  is  taken  into  account,  the  potential  energy  U  becomes  U  —  R, 
where  R  is  the  disturbing  function  and  is  expressed  by  (§  23) 


Hence  H  becomes  H0  —  R  and  consequently  by  §  127  the  constants  of  the 
approximate  problem  are  in  the  more  complete  problem  subject  to  variations 
which  are  denned  by  the  equations 

dar  _     dR       d/3r  _     dR 
~dt  =  ~Wr'     ~dt'=   +dar' 

Here  R  is  supposed  to  be  expressed  in  terms  of  the  constants  mentioned  in 
§  136,  which  refer  to  the  motion  of  the  planet  considered  undisturbed,  and 
the  time  as  it  occurs  in  the  expression  of  the  coordinates  of  the  disturbing 
planets.  When  instead  of  the  canonical  constants  arising  in  the  solution  of 
the  Hamilton-Jacobi  equation  the  ordinary  elements  of  elliptic  motion  are 
employed,  the  equations  for  the  variations  are  no  longer  of  the  above  simple 
type,  but  take  the  more  complicated  form 

-  -  S  \A     A\^— 

'  1    r>     '* 


dt 

where  Ar  represents  any  one  of  such  elements.     Since  we  have  found  the 
expressions  for  all  the  Poisson's  brackets,  the  equations  for  the  variation  of 


138-140]  Variation  of  Elements  147 

the  usual  elliptic  elements  can  at  once  be  written  down  in  an  explicit  form. 
They  are  as  follows  : 

da  i —    dR 

dt  =2Va/^.^- 

de  cot  d>   dR     tan  ^d>  cos  d>    dR 

dt  *J  lL(t       9'57 


di  1  dR         tan|i 


'•N  •  ~"        ^ 


dt          cos  d>  sin  i  V/*a  8fl      cos  d>  v  yu.a    WOT      de 

dn=  1  dR 

dt      cos  d>  sin  i  V/ta     9* 

rfor  _  cot  d>  9.B         tan  \i      dR 

~~TT   —      / -    •  "^        h   •••          ~~ ' "/•——•  •  "jr7" 

rtc       v  yLia     c*e      cos  d)  v/ia    0* 

rfe  97iJ      tan  ^d>  cos  d>    dR          tan  ^i        dR 

dt  i  r"  da  v/xa  de      cosd>.v/Lta-    ^* 

A  slight  simplification  has  been- made  by  writing  sin  d>  in  place  of  e  in  the 
coefficients  of  the  partial  differentials  of  R. 

140.  The  above  set  of  equations  for  the  variations  of  the  elements  is 
fundamental.  An  important  point  must  be  noticed  in  regard  to  them.  The 
variation  of  a  entails  a  corresponding  variation  of  n  which  is  determined  by 
the  relation  n2as  =  p.  Now  the  disturbing  function  R  is  a  periodic  function 
of  the  mean  anomaly  and  is  expressed  in  terms  of  circular  functions  of  mul- 
tiples of  nt.  Hence  the  derivative  of  R  with  respect  to  a  would  contain  the 
same  circular  functions  multiplied  by  t  and  this  introduction  of  terms  not 
purely  periodic  would  be  inconvenient.  The  difficulty  is  avoided  by  an 
artifice  which  should  be  carefully  noted. 

We  consider  n  (as  distinct  from  a)  to  occur  only  in  the  arguments  of  these 
periodic  terms.  Otherwise  a  is  used  explicitly  or  if  it  is  more  convenient  to 
use  n  outside  the  arguments,  n  is  simply  a  function  of  a  given  by  n2a3  =  p. 
Now  e  enters  into  R  only  in  the  form  nt  +  e  through  the  mean  anomaly, 
so  that 

dR  _  1  /9.R\ 

8e      1,  \97i/a=collst.'        , 
Hence 

-  =  -2V^T  — 

dt  "da 

2\  dn-/dR\ 


dR\  dn dR 

.9a/w=const.        da  8e 

^9^\  dn  da 

~^  *  ~j—  •  ~T7  -\-  . . . 

<  f'a/w=con8t.  U/a     Ml 

10—2 


148  Variation  of  Elements  [CH.  xm 


or 

de      .  dn          -.    /— 7—  /9jR\ 
-r  +  t  -j-  =  —  2  va/fjb  (  ^-  ] 

(It  Ctt  .  \U(l  /  )i  =  const. 

If  then  we  take  e'  instead  of  e,  where 

de        dn  _  de' 
dt        dt      dt 
or 

e  +  nt  =  e  +  Indt 


=  e'  +  I 


the  form  of  the  above  equations  for  the  variations  of  the  six  elements  will  be 

unaltered,  since 

dR  =  dR 

de  ~  de' 

but   their  natural  meaning  will  be  so  far  altered  that  (1)  n  in  the  mean 
anomaly  is  not  to  be  varied  in  forming  the  derivative  with  respect  to  a,  and 

(2)  nt  in  the  mean  anomaly  is  to  be  replaced  by  Indt.     The  secular  terms 

which  would  arise  from  the  cause  mentioned  are  thus  avoided. 

The  value  of  n  is  deduced  directly  from  the  value  of  a,  and  we  have 


4  f 

=  ^ 


- 

a    -  dt. 


If  this  integral  be  denoted  by  p  we  have  also 

d*p  .—,-   da         3   dR 


or 


which  gives  the  finite  variation  of  this  part  of  the  mean  longitude  in  the 
disturbed  orbit. 

141.  When  e  (and  therefore  <f>)  is  small,  and  this  is  commonly  the  case, 
the  coefficients  in  the  variations  of  e  and  OT  which  contain  cot  (f)  as  a  factor 
become  large.  This  gives  rise  to  a  difficulty  which  can  be  avoided  by  intro- 
ducing the  transformation 

li-i  =  e  sin  -or,     &j  =  e  cos  ta. 

The  result  of  making  this  change,  which  can  be  verified  without  difficulty,  is 
to  substitute  for  the  corresponding  pair  of  equations 

dhi         cos  <£  9-R       &!  tan  |t    dR          h^  cos  <£        dR 


dt  V/ia  9&!  cos  <£  V/ia  di  2  cos2  1^>  V/^a  9e 
dki_  cos<£  dR  Ajtan^i  9^  ^  cos  <f>  dR 
'dt  Via  9A,  cos  <  x/tta  di  2  cos2<>  Via  9e 


140-142]  Variation  of  Elements  149 

Similarly,  when  the  angle  between  the  plane  of  the  orbit  and  the  plane  of 
reference  is  small,  a  pair  of  coefficients  in  the  variations  of  i  and  fi  become 
large,  and  the  transformation 

h-z  =  sin  i  sin  H,     kz  =  sin  i  cos  U 

is  useful.  The  result,  which  can  be  verified  with  equal  ease,  is  to  replace 
the  equations  named  by  the  pair 

rf//2  cos  i        dR  h2  cos  i  /dR     dR\ 


dt         cos  <f>  V/^a  dk.2     2  cos2^'  cos  <f>  \//JM    \9w      9e  / 

dk2_          cosi       dR  k2cosi  fdR     dll 

dt         coadtvua  dh.2     2cos2Aicosd>  \fua  V?w      9e 


142.  Another  form  of  the  equations  for  the  variations  of  the  elements, 
in  which  the  disturbing  forces  appear  explicitly,  is  of  great  importance.  Let 
8,  T  be  the  components  of  these  forces  in  the  plane  of  the  orbit  along  the 
radius  vector  and  perpendicular  to  it,  and  W  the  component  normal  to  the 
plane.  Let  u  be  the  argument  of  latitude  and  (X,  /u.,  v)  the  direction  cosines 
of  the  radius  vector,  so  that  (§  65) 

A.  =  cos  u  cos  fl  —  sin  u  sin  II  cos  i 
fjb  =  cos  u  sin  fi  +  sin  u  cos  fl  cos  i 
v  =  sin  u  sin  i. 

The  direction  cosines  of  the  transversal  and  of  the  normal  to  the  plane  may 

be  written 

9X     9/4      dv       ,      1     9\        19/4        1     dv 
du'    du'    du  sin  u  di  '    sin  u  di  '    sin  u  di 

which  must  satisfy  the  conditions 


• 

duj    '  sin'u 


If  <r  be  any  one  of  the  elliptic  elements,  we  have  also 

dR_dR  dx     dR  ty     dR  fa 
da-      dx  'da-      dy  '  da-      dz  'da' 

But  the  component  of  the  disturbing  forces  along  the  axis  of  x  is 


_        —  /v^-     r  r\     •*•     j       •  o  • 

ox  du         sin  u  01 

Hence 


sin  u 


-  — 

o       •   *  •*  ••  1        'o          >      •  o-'o 

da-  \du  da-/      smu      \di    da 


150  Variation  of  Elements  [CH.  xm 

by  the  conditions  mentioned.     Now 

r  =  a(l  —  ecosE),         tan  -Jw  *= 


u  =  iff  —  O  4-  iv,        E  —  e  sin  E  =  nt  +  e  —  OT. 

In  accordance  with  §  140  we  treat  n,  as  it  occurs  implicitly  in  u,  as  inde- 
pendent of  a,  and  replace  nt  by  Indt. 

Hence 

dR      08r'    rS 
~o~  =  £>  ^5—  =  — 
da          da      a 


di      sinw 


duj      sin  u      di 
(since  X  contains  O  both  explicitly  and  implicitly  through  u) 

=  rT]2  (  ~- 3-7^  )  —  1  [  +  -     ~^(~~n) 

rW 

=  rT  (cos  i  —  1 )  4-  — —  (-  sin  u  cos  u  sin  i) 
sm  wv 

=  —  2rT  sin2  %i  —  r  W  cos  u  sin  *'. 

The  remaining  elements  enter  into  (X,  //-,  v)  only  implicitly  through  w,  so 
that  in  their  case 

^  _  <?  <*L  _L    r?  f8-Y   ?^  4-  r^  *  ^9X  9X^  9w 

«  O  r\       T  »  J  A  I  S      J'o         i 


3o-         80-  \&M/     3o-      sin  M      \  di '  duj  3 


W  ^  fd\  8X\ 

2,  I  TTT  .  — 

in  w      \  m    ou] 


)(T 


Hence 

3.R      Q,         .     j-j  dE        ™  9w  3£^ 
"37"  "^  9e  dE'^e 

=  S .  a2  e  sin  J£/r  +  aT  sin  w/sin  ^ 
=  aS  tan  </>  sin  w  +  aT  sec  0  (1  +  e  cos  w). 
Since  r  and  w  are  both  functions  of  e  —  OT, 


142]  Variation  of  Elements  151 

and  finally 


dR  _  Qdr  dw 
•^  --  *3  ^  —  r  I  J-  ~zr~ 
de  oe  oe 


sn  w 


---  „ 
sin  E 

esirfE   \  1  -  1 


„(  esirfE   \  1  -  1    \ 

=  aS   -  cos  E  +  -  -~  }  +  rrsra  «n  -—        —=,  +  ..  -  = 

V  1  —  e  cos  E)  \l-ecosE     1  —  e2/ 

„,     e  —  cosE  fl+ecosw         1 


.=—        —  0  ,     —  —     ^  -  :, 

1  —  e  cos  E  \l-e-          1  —  e' 

=  —aS  cos  w;  +  rT  sin  w  (2  +  e  cos  w)  sec2  0. 

It  only  remains  to  carry  the  expressions  found  for  the  derivatives  of  R  into 
the  equations  of  §  139  for  the  variations  of  the  elements.  The  results  are  as 
follows  : 

da 

-^7=2  Va'/A*  {&  tan  </>  sin  w  +  T  sec  (j>  (1  +  e  cos  w)] 

^e          _ 

—    =  v  a/  p  cos  <|>  (>S  sin  w  +  T(cos  w  +  cos  E)} 

'-    =  rW  cos  M/COS  rf>  Vtta 
rti 

c?O  .     . 

-^-  =  r  W  sin  u/cos  9  sin  i 


2  sin2i</>    ~  +  2  cos  0  sin'it. 

From  the  first  two  equations  we  get  for  the  variation  of  the  parameter 
ja  =  a  (1  -  e-) 

^  =  cos2  6  ~  —  2a  sin  <f>  -y-  =  2?-  T  cos  0  v'a/u  . 
cU  dt  at 

It  has  been  convenient  to  derive  the  above  important  set  of  equations  from 
those  which  involve  the  derivatives  of  the  disturbing  function.  But  their 
form  would  be  the  same  if  the  components  of  the  forces  were  not  such  as  can 
be  expressed  as  the  differentials  of  a  single  function.  Thus  they  hold,  for 
example,  in  the  case  of  elliptic  motion  disturbed  by  a  resisting  medium. 

Since  w2as=/u,  is  constant,  the  equation  for  "the  variation  of  a  maybe 
replaced  by 

~  =  —  3  {$sin  <£sin  w  +  T  (I  +  ecosw)}/acos<f>. 
lit 


1  52  Variation  of  Elements  [OH.  xm 


Also 


-  (e  —  •BT)  =  —  2rSf^/(/jui)  —  cos  </>  --  +  r  W  sin  w  t^an  ^i 


=  {(a  cos2  </>  cos  w  —  2r  sin  <£)  S  —  rT  sin  w  (2  +  e  cos  w)}/sin 
which  gives  the  variation  of  the  mean  anomaly, 

dM      d  ,  fdn  , 


part  of  the  variation  of  nt  being  included  in  e  as  explained  in  §  140  and 
mentioned  above. 

143.  It  has  been  seen  in  §  139  how  the  canonical  solution  of  the  problem 
of  undisturbed  elliptic  motion  leads  to  the  canonical  equations  appropriate  to 
the  form  of  motion  which  follows  from  the  introduction  of  disturbing  forces. 
With  a  slight  change  of  notation, 

L  =  a  =  V(/K»),  I  -  nt  — 13  =  e  -  «r  +  nt 


H-«%  0=       -*-«r-U 

JJ=  «3  =  Vl/ua  (1  -  e2)}  cos  i,     A  =      -  /33  =  fl 

and  the  canonical  equations  become 

dL  ^dR       dl  =  _dR 

dt  ~~  dl  '      dt~     dL 

dG  _dR       dg _     dR 
~dt  ~dg'     dt  =  ~dG 

dH_dR       dh_     dR 
dt~dh'      'dt        dH' 

But  there  is  here  a  change  in  the  meaning  of  R  due  to  replacing  the  element 
—  /3  by  the  mean  anomaly  I.  If  the  disturbing  function  in  the  usual  form 
quoted  in  §  139  be  denoted  by  R0,  the  variation  of  I  follows  from 


d  .,        .  _      9jR0      dR  _  9^o 
dt(l~  ~dL'    dL=dL~ 

and  therefore 

ndL  =  R0-  L'L-^dL  =  R0 


This  change  in  R  has  no  effect  in  the  other  equations,  and  since  R  is  a 
function  of  e  —  OT  +  nt,  dR/dl  is  the  same  thing  as  —  dR/d/3.  The  above 
canonical  equations  are  precisely  those  on  which  Delaunay's  theory  of  the 
Moon  is  based. 

Without  changing  L  let  the  transformation 

L  —  G  =   },     G  —  H  =  .?,    —  c  —  /<  =  &>,      —h  =  co2, 


142-144]  Variation  of  Elements  153 

be  made.     Then 

\dL  +  co1dp1  +  a)zdpz  —  (IdL  +  gdG  +  hdH)  =  0 

and  this  expression  is  therefore  a  perfect  differential.  Hence  by  §  125  the 
transformation  from  the  variables 

L,  G,H;   l,g,h 
to  the  variables 

L,  /»i,  /?2;    X,  ft>!,  a>2 

is  one  which  leaves  the  equations  of  motion  in  the  canonical  form.  The 
angle  A,  =  e  +  nt  is  the  mean  longitude,  and  o)1  =  -  OT,  &>2  =  —  ft  are  the  longi- 
tudes of  perihelion  and  the  node,  reversed  in  sign. 

Again,  consider  the  transformation 

%  =  (2p)^  cos  &),     ?;  =  (2p)2  sin  o>. 
In  this  case 

r)dj;  —  wdp  —  —  2p  sin2  codco  +  sin  o>  coswdp  —  wdp 

sin  2ft>  —  &) 


is  a  perfect  differential.     Hence  the  variables  L,  pl,  p.2;  \,  (o1}  eo2  can  be 
changed  to 

L,  &,  %z\  \  i7lt  % 

and   the   canonical  form   of  the  equations  will  still  be  preserved.     These 
variables  have  been  used  extensively  by  Poincare.     Since 


(sin^>  =  e),  |1}  7?!  are  of  the  order  of  the  eccentricity,  and  are  called  by  him 
the  eccentric  variables.     Similarly,  since 


sn2 

£2,  rj2  are  of  the  same  order  as  the  inclination,  and  are  therefore  called  the 
oblique  variables. 

144.  The  account  which  will  be  given  of  the  lunar  theory  in  later 
chapters  will  be  based  on  a  method  which  is  quite  different  from  Delaunay's. 
But  the  latter  is  in  reality  very  general  and  therefore  Delaunay's  mode  of 
integrating  the  canonical  equations  of  the  previous  section  will  now  be 
indicated.  The  form  of  the  disturbing  function  will  be  taken  to  be 

R  =  —  B  —  A  cos  (i-i  I  +  i2g  +  i3h  +  itrit  +  q)  +  R^ 
=  -B-Acos0  +  R1  =  R0  +  R1 

where  Rl  represents  an  aggregate  of  periodic  terms  similar  to  the  one  written 
down  and  n',  q  are  constants.  The  term  B  and  the  coefficients  A  are 
functions  of  L,  G,  H  only  and  in  comparison  with  B  these  coefficients  are 
small  quantities  of  definite  orders.  Let 

#!  =  il  I  +  i*g  +  iji  —  6  —  i4n't  —  q. 


154  Variation  of  Element*  [CH.  xm 

Then  the  variables 

L,G,H-l,g,k 

can  be  replaced  by 

L,G',H';  ir^frh. 

provided 


is  a  perfect  differential  ;  and  this  condition  is  clearly  satisfied  if 
G'  =  G  -  ir*i*L,     H'  =  H-  irlizL 

for  then  d  W  =  0.     If  now  -Ra  =  0,  a  solution  of  the  problem  can  be  found. 
For  corresponding  to  the  equation 

R  =  -  B  -  A  cos  (#!  +  i4nt  +  q) 
the  Hamilton-Jacobi  equation  takes  the  form 

/.  dW  \     dW 

-B-Acos(i1^r  +  itn't  +  gr   +  —  -  =  0 

\      oLi  J         Ot 

and  a  solution  involving  three  constants  G,  g,  h'  is 

W  =  Ct  +  ir1  1  OdL  -  i~lL  (itn't  +  q)  +  g'G'  +  h'H' 

provided 

-B-A  cos  0  +  C-  i~lL  .  itn'  =  0. 

This  equation,  which  is  in  fact  one  integral,  may  be  written 
C  =  B1+Aco&0,     B,  =  B  +  i.n  ,  i^L.  . 
The  solution,  by  §  126,  takes  the  form  (ar  =  C,  g',  h'  ;  &r  =  c,  -  G',  -  H') 

t+c+  ir1  JL  \OdL  =  0,     ir1  Ol  =  ir1  (O-i.n't-q) 

du  J 

a 


G'  =  const.,  g  =  g'  +  il-l^-,\0dL 

H'  =  const.,  h  =  h'  +  ir1  555  f^^- 

0/2    J 


The  lower  limit  of  the  integral  involved  is  a  function  of  (7,  G',  H',  but  the 
integral  is  so  defined  that  the  integrand  6  vanishes  at  this  limit.  The 
solution  can  also  be  written 


~ 


1 14,  145]  Variation  of  Elements  155 

At  this  point  (C,  g',  h' ' ;  c,  —  G' ,  —  H'}  are  absolute  constants,  resulting  from 
the  solution  of  a  Hamilton-Jacobi  equation  when  the  Hamiltonian  function  is 
R  —  Rl.  Hence,  by  §  127,  the  further  treatment  of  the  problem  depends  on 
taking  these  constants  as  new  variables,  and  solving  the  canonical  system 

dC        dR,     dGT        dR, 


dt          8c  '     dt.        dg'  '     dt          dh' 

dt=~dC'    ~dt=~"dG"     ~dt  =  ~d~H'' 

But  circumstances  now  arise  which  require  further  examination.  For  Rl  is 
now  a  function  of  the  new  variables,  instead  of  the  old,  and  the  form  of  the 
function  is  important. 

145.     In  the  partial  solution 

C  =  B,  +  A  cos  0,     ^  =  V  {A*  -(C-  B,Y-}  =  Asin0 

where  Bl)  A  are  functions  of  ©  (and  the  constants  C,  G',  H'),  and  @,  0  are 
functions  of  t  to  be  determined.  The  forms  to  be  expected  may  be  seen  in 
this  way.  The  above  equations  give 

<H)=/(cos0),      -/'(cos  6}~  =  A 
and  therefore 

t+c=l(f>  (cos  0)  d6  =  0/00  +  2tfr  sin  r0 


when  0  vanishes  with  t  +  c.     Hence  0  —  00  (t  +  c)  is  an  odd  periodic  function 
of  0  and  therefore  of  X  =  00  (t  +  c).     Thus,  00  being  some  constant, 

0  =  X  +  20,  sin  r\,     \  =  60(t  +  c) 
and 

©  =/(cos  0)  =  @0  +  S@r  cos  r\. 

These  forms,  which  without  a  critical  examination  of  the  conditions  have 
only  been  made  plausible,  are  actually  found  in  practice.     It  follows  that 

L  =  t\  @0  +  i^&rcos  r\,  G=G'+i.2®0+i2'2 


,        90    A  sin  0  ,.  «•      • 

9  =  9  +  \^7^-  —a  —  d\  =  g  +g0(t+c)  +  Zgr  sin  r\ 

OLr  (70 

30    A  sin  0  , 
fj-,  .—          ,d\  =  h'  +  h0  (t  +  c}  +  ±hr  sin  r\ 


/" 
J 


and  the  original  variable  I  is  given  by 
1^1  =  6  —  iji't  —  q  —  i2g  —  iji 

=  \  —  i^n't-q  —  ii  {g'+gQ(t  +  c)}  —i3  {h'  +  ha(t+c)} 


156  Variation  of  Elements  [CH.  xin 

Now,  since  0  and  (•)  contain  G,  0',  H',  these  constants  also  enter  into  g0,  h0 
and  therefore  into  the  coefficients  of  t  in  the  arguments  of  the  terms  in  R^. 
Hence  t  will  appear  outside  the  circular  functions  in  the  derivatives  of  R1 
with  respect  to  C,  G',  H'.  This  inconvenient  circumstance  must  be  avoided 
by  a  change  of  variables.  Now 


d    6  d®  =  6d®  -(t  +  c)dC+(g-  g')  dG'  +  (h  -  h')  dH' 
by  the  form  of  the  partial  solution,  and  therefore 

d(ct-j®d0\=-®de-cdC  +  (g-  g')  dG'  +  (h  -  h')  dH'  +  Cdt. 

This  is  a  perfect  differential  and  when  each  side  is  expanded  in  the  form  of 
a  secular  and  a  periodic  part,  the  same  must  clearly  hold  true  for  each  part 
separately,  at  least  when  the  number  of  periodic  terms  is  finite  ;  and  in 
practice  the  remainder  after  a  certain  number  of  terms  must  be  treated  as 
negligible.  But 

cos  rX) 


o 

=  A0  +  SAr  cos  r\,     A0  =  ®0  +  £2r  ®r0,. 
Hence,  when  the  periodic  terms  are  omitted, 

Cdt  -  A0d\  -  cdC  +  g0(t  +  c)  dG'  +  h0(t  +  c}  dH' 

is  a  perfect  differential,  to  which  d  (A0X)  may  be  added  ;  and  therefore  the 
variables 

C,  G',H';  c,g',h' 
can  be  replaced  by 

A0,  G',  H'  ;  X,  K,  rj 
where 

K  =  9'  +  9o  (t  +  c),     77  =  h'  +  h0(t  +  c). 

This  follows  from  §  125,  which  shows  that  at  the  eame  time  RI  must  be 
replaced  by  .Rj  —  C.  All  is  now  expressed  in  terms  of  the  last  set  of  variables, 
and  secular  terms  are  thus  removed  from  the  arguments  of  the  terms  in  R^. 

It  is  convenient  to  make  a  final  simple  transformation.     Since 

(ijX'  —  X)  dAn  4-  ivtcdAo  +  irfdA.0  =  —  d  {A0  (i4nt  +  q)}  +  i^n'Andt 
if 

.  ijX'  =  X  —  i2K  —  i;?;  —  i4n't  —  q 
the  variables 

A0,  G',  H'  ;  X,  K,  r) 
can  be  replaced  by 

.A'  =  1^0,  G"  =  G'  +  i2A0,  H"  =  H'  +  i,A0;  V,  /e,iy 

but  at  the  same  time  it  is  necessary  to  add  itn'AQ  to  R^  —  C.     Thus  finally,  if 

R'  =  R,-  C  +  i4n'A0 


145,  146]  Variation  of  Elements  157 

the  system  of  canonical  equations 

dA'        aft      d&"        d&     dH"        dR' 


dt~       ax'  '      dt    '        dtc  '      dt  drf 

d\'==_aB/        d*  =  _dR/       dr,       _dR' 
dt  ~     8A"       dt'~     dG"'     dt  ~~      dH" 
is  obtained. 

146.  If  the  value  of  X'  be  compared  with  the  expression  for  I  in  terms  of 
X  it  will  now  be  seen  that 

i^l  =  ^x,'  +  2  (0,.  —  i.2gr  —  ijir)  sin  ?-X 

and  thus  X'  and  I  differ  only  by  periodic  terms.  The  same  is  true  of  K,  g  and 
77,  h.  The  periodic  terms  would  disappear  with  A,  as  also  those  in  ©  and  6, 
and  A0  would  coincide  with  00  and  ®.  Hence  the  final  variables  are  the 
same  as  the  original  variables  when  .4  =  0.  The  form  of  R'  differs  from  that 
of  R  mainly  in  the  complete  removal  of  the  term  A  cos  6,  and  naturally  the 
most  important  term  will  be  first  selected  for  elimination.  Periodic  terms 
will  be  introduced  into  the  arguments  of  R',  but  it  is  easily  seen  that  on 
expansion  they  give  rise  to  periodic  terms  of  a  higher  order  than  A  cos  6. 

The  same  process  can  be  repeated  indefinitely,  until  all  sensible  terms  are 
one  by  one  removed,  together  with  those  of  a  higher  order  introduced  at  an 
earlier  stage.  It  has  been  assumed  that  ^  is  not  zero.  If  ^  =  0,  i^g  or  i3h 
can  take  the  place  of  ij  1.  There  are  also  terms  for  which  ^  =  i2  =  iz  =  0.  In 
the  lunar  problem  these  depend  on  the  mean  longitude  of  the  Sun  and  are 
removed  by  a  single  preliminary  operation  analogous  to  the  above. 

Delaunay's  expression  for  the  disturbing  function  contains  over  300 
periodic  terms,  and  their  removal  involves  practically  500  operations  of  the 
above  kind,  reduced  to  the  application  of  a  set  of  formal  rules.  This 
immensely  laborious  task  was  carried  out  unaided.  But  the  result  is  the 
most  perfect  analytical  solution  which  has  yet  been  found  for  the  satellite 
type  of  motion  in  the  problem  of  three  bodies.  The  solution  is  not  limited 
to  the  actual  case  of  the  Moon,  since  it  is  expressed  in  general  algebraic 
terms.  The  satellite  type  of  motion  may  indeed  be  defined  as  that  type  for 
which  the  Delaunay  expansions  are  valid.  It  seems  an  interesting  problem 
of  the  future  whether  such  satellites  as  Jupiter  VIII  and  IX  will  be  found 
to  satisfy  this  definition.  Their  conditions  differ  widely  from  those  of  the 
lunar  problem,  in  particular  in  the  fact  that  the  motions  are  retrograde. 


CHAPTER   XIV 

THE    DISTURBING   FUNCTION 

147.  The  development  of  the  disturbing  function  R  in  a  suitable  form 
gives  rise  to  many  difficulties,  partly  of  analysis,  partly  of  practical  computa- 
tion, and  is  the  subject  of  an  extensive  literature*.  It  is  possible  to  deal 
here  only  with  a  few  of  the  more  important  points. 

The  principal  part  of  the  disturbing  function  for  two  planets  involves  the 
expansion  of  A"1,  the  reciprocal  of  their  mutual  distance.  It  is  therefore 
important  to  consider  the  nature  of  this  expansion,  or  rather  of  A~2*  in 
general,  where  s  is  half  an  odd  integer.  For  this  more  general  form  will 
give  the  derivatives  of  A"1,  A2  being  a  rational  quantity,  and  these  will 
naturally  occur  when  A"1  is  expanded  in  terms  of  any  contained  parameter. 

It  is  convenient  to  consider  first  the  case  of  two  circular,  coplanar  orbits. 
Then,  if  H  is  the  difference  of  longitude  in  the  plane, 

A2  =  aj2  -f  tt22  -  2a:  02  cos  H 
at ,  a2  being  the  radii  of  the  orbits.     Let 

aj  <  a2,     a  =  «i/a2,     ^H  =  log  z,     i?  =  —  1 
and  therefore 

tt2~2  A2  =  1  +  a2  -  2a  cos  H  =  (1  -  az)  (1  -  a*"1). 

Hence  the  function  to  be  examined  is 


=  (1  +  a2  -  2a  cos  H)~g  =  $bs°  +  2  6,*  cos  iH. 

i 

Since  the  function  is  unaltered  when  z  and  z~l  are  interchanged,  bs~i  =  bgi, 
and  i  may  be  treated  as  positive.  The  coefficients  b8{  are  called  Laplace's 
coefficients.  By  Fourier's  theorem, 

&/  =  —    (1  —  oz)~8  (1  —  as"1)"*  ^-1  dz 

7TI  J 

=  -  I    (1  +  a2  -  2a  cos  t)~'  cos  t«d< 

W.'O 

*  Cf.  H.  v.  Zeipel,  Encykl.  der  Math.  Wist.,  vi,  2,  pp.  560-665. 


147,  us]  The  Disturbing  Function  159 

The  first  (complex)  integral  is  due  to  Cauchy ;  the  path  of  integration  is 
taken  round  a  circle  of  unit  radius.      By  introducing  the  Weierstrassian 

elliptic  function 

p  0)  =  z  -  i  (a  +  a-1) 

Cauchy's  integral  clearly  becomes  an  elliptic  function,  and  Poincare  has 
shown  how  this  function  can  be  reduced  to  a  calculable  form.  But  another 
method  will  be  followed  here. 

The  coefficients  bsi  are  easily  developed  as  power  series  in  a-.     For,  with 
the  use  of  gamma  functions, 

(1  -  az}-*  (1  -  or-1)-  -  2       r^-  +-P).-  a?**  .  2      r^+gl_    0.1  z~i 
pT( 

and  therefore,  when  p  =  q  +  i, 


r(<7  +  i) 


a*,  2, 

<7 


But  this  can  be  recognized  as  a    hypergeometric  series,  and  when   it   is 
expressed  in  the  ordinary  notation, 

bj=^F(s,  s  +  i,  t  +  1,  ^  ...............  (2) 


By  the  known  properties  of  the  hypergeometric  series,  this  expansion  is 
convergent  when  a<  1.  There  are  many  equivalent  forms,  but  (2)  is  enough 
for  the  present  purpose. 

148.  Laplace's  coefficients  are  subject  to  several  formulae  of  recurrence, 
which  facilitate  their  calculation.  That  such  exist  follows  from  the  known 
relations  between  sets  of  three  contiguous  hypergeometric  functions.  Instead 
of  finding  them  directly,  a  more  general  function 


may  be  considered,  for  this  reduces  to  bsi  when  j  =  0.     In  the  integral  0 ) 
write  z  =  a%,  and  then 


It  follows  that 


TOO 

The  equivalent  forms 


TTICC 


I  <!  - ' 


160  The  Disturbing  Function  [CH.  xiv 

show  at  once  that 

Wj«#-Ji/+»_*JM«  .....................  (3) 


Again, 


=  (1-  a.zQ-s-i  (I  -%-*)-*  {(s-i-l 

When  these  expressions  are  integrated  along  a  path  lying  between  the  limits 
1  <  |  f  <  or2,  where  the  functions  are  regular,  the  first  integrand  returns  to 
its  original  value.  Therefore 

(i  -  s  +  1)  a  Bgt+1>s  -  (i  +j  +  *'a2)  £/'•'  +  (i  +j  +  s  -  1)  a  B^  =  0  .  .  .(4) 
The  identity 

(i-a^-^a-r1)"*^'1 

=  (1  -'aa£)—  •'-'  (1  -  t-1)-"-1  {(I  +  a*)  f'+J-1  -  a2fw-  £'+j~2} 
gives  similarly  on  integration 


(s  +  j)  BiJ  =  s(l  +  a2)  5*'^  -  8CLBi+*'j  -  sa  Bll\j 

^        J'      s  '       s+1  s+l  s  +  1 

and  after  eliminating  the  last  term  by  means  of  (4)  with  s  +  1  in  the  place  of  s, 


When  j  =  0,  (4)  and  (5)  give  formulae  which  apply  to  Laplace's  coefficients. 
Derivatives  of  the  latter  with  respect  to  a  can  then  be  expressed  as  linear 
functions  of  Bj'J. 

i 

149.  Newcomb's  method  of  calculating  the  coefficients  bsi,  together  with 
their  derivatives  in  the  form  subsequently  required,  can  now  be  explained. 
Let 


2s  =  n,     8  =  -,     D  =  a 

dy?  da 


and  let 


This  is  not  Newcomb's  definition  of  c^i,  but  it  is  the  equivalent.     Thus 

D  cji  =  {  J  (»  -  1)  +  i  +  2j}  cj'l  +  cn^+1 
and  therefore 

J+1  ............  (6) 


so  that  these  derivatives  of  a  higher  order  are  easily  deduced  from  those  of 
the  next  lower  order.     Let 


148-iso]  The  Disturbing  Function  161 

and  then,  by  (4), 


where 


The  development  is  to  be  carried  to  a  definite  order  fixed  by  i  =  k,  say  11. 
In  the  first  place  pnk>J  is  calculated  for  the  required  values  of  n,  j  by  a  direct 
method.  Next  pnk~l'i,  •••,  pnl'j  are  deduced  in  succession  by  (7).  For  i  =  I, 
s  =  -|-,  the  formula  (3)  becomes 


+      a  dl'   = 
or 


The  first  coefficient  d0'  °  is  calculated  directly.    Then  (8)  gives  d0'  *  (  j  =  1,  2,  .  .  .  ) 
in  succession.     The  formula  (5),  when  i  =  0,  gives 


\n  [in  +  (j  +  in)  a2]  c^2  -  $n  (j  +  n)  a  cj 
or  • 

(),j  _  w      2    /       w 


c"+!   i»  H»  +  (j  +  i«)  <f]  -  i»  (j + «)  «P;;' 


(9) 

t+2 

whence  cn°'J  (n  =  3,  5,  ...)  are  found  in  succession.  It  only  remains  to  form 
Cni'j=pni'^cni~1'^ ]  (i  =  1,  2,  ...)  and  the  calculation  is  then  complete.  The 
successive  derivatives  are  finally  derived  by  the  use  of  (6). 

The  employment  of  a  chain  of  recurrence  formulae  in  practical  computa- 
tions requires  care,  because  they  are  apt  to  involve  an  accumulation  of 
numerical  error.  It  is  the  merit  of  Newcomb's  method  here  described  that 
it  is  not  only  simple  but  very  accurate. 

150.  The  quantities  which  must  be  calculated  directly  are  d0'0  a,ndpnk>i, 
where  n—1,  3,  ...,  j  =  0,  1,  2,  ...,  and  k  is  the  highest  value  of  i  to  which  the 
expansion  is  carried.  Now 


a  complete  elliptic  integral  which  can  be  found  in  »  great  variety  of  ways. 
Newcomb  commends  for  the  purpose  the  arithmetic-geometric  mean,  which 
follows  from  the  identity 

/•I"-  /•£*• 

(an2  cos2  </>  +  V  sin2  $)~  *  <&  =>        (a?n+l  cos2  ^  +  62,l+1  sin2  ^r)~  *  dtyr 
J  o  7o 

where 

2ft)i+i  —  dn  +  On>      6"n+i  =  ft»0n< 
P.  D.  A.  11 


162  The  Disturbing  Function  [CH.  xiv 

This  is  obtained  immediately  by  the  transformation  of  Gauss 

.      ,_  2(/nsin^r 

(an  +  bn)  cos-  ^  +  2aw  sin2  ^ 

and  can  be  extended  indefinitely  by  successive  steps.  It  is  obvious  that  the 
sequences  an,  bn  have  a  common  limit  A  and  hence  that  the  value  of  the 
integral  is  7T/2A.  In  the  present  case 

<f,  =  l-a,     &!  =  !+«,     c,0'°  =  24-1 
and  this  indicates  one  way  in  which  c-?>  °  is  easily  obtained. 

The  calculation  of  pnk>i  is  based  on  the  hypergeometric  series  (2).     It  is 
clear  that 


s  +  i,  i  +  1,  a2)  =  ^(«  +  1,  s  +  i+  1,  i  +  2,  a2) 

i  +  1 

and  therefore  generally 

i>+j)  r(«+t+j)    r<»  +  i) 
Wfo...)-  -r^)--r(,  +  i)    -r(i+j  +  l)f 

Hence,  by  (2),  0 


-  9QitT/     ,    ;    „,-•,•   ?"   ,    ;  i    i     aa 

-^" 


and  therefore,  since  n  =  2s, 

f.j=    B's3    =^n  +  i+j-l   F  (±n  +j,  jn  +  i  +j,  i  +j  +  1,  a2)  g 
F»   '    jg»'~1'J  i+j          ']^(^n+j,  ^w  +  i+j-1,  t'+j,  a2) 

5 

The  quotient  of  .the  two  hypergeometric  series  can  be  converted  into  a 
continued  fraction  by  a  known  theorem*  of  Gauss,  and  as  it  converges 
rapidly  a  few  terms  suffice  to  give  its  value.  By  this  method  Newcomb 
determined  the  required  values  of  pnk'i. 

151.  In  order  to  obtain  the  desired  form  of  the  continued  fraction  it 
is  not  necessary  to  introduce  the  hypergeometric  series.  By  (3)  and  the 
following  equation, 

Bi+i,j         aBi+l>j+1^Bl'j+l 

1    >J         *'  '"'  * 


'  AJ'""  _  g-  _          S  8 

Pn-2     ~~    ni-l,j  +  2  ~   D«77+l  _Ja  D*-1.J+1 
s  —  1  s  s 

and  by  (4), 

(i  -s  +  1)  afl;+w+1-(i  +j  -f-  1  +  ia2)  tii;j+1+  (i  +j  +  s)  a  B^1'^1  =  0. 

*lChrystal's  Algebra,  n,  p.  495. 


150-152]  The  Disturbing  Function  163 

These  are  three  linear  equations  in  Bsi+1»J+1t  5/-J'+1,  Bj-*'J+l,  which  can  be 
eliminated.     The  result  may  be  expressed  in  the  form  : 

(i-s+l)a     i+j+itf+l     (i+j  +  s)a  ;  =  0. 


—  2  -t^n  -  2 

1*  +  1,  /  i  +  lj.7 

+  CiPn  Pn 

After  expansion  and  division  by  (1  —  a2)  this  gives 


or 


1'  ^  ~  ia}  i<*  -  *  +  !)  «X£  "  -  (»'  +  J  +  1)1 
Therefore  (7)  gives  (2s  =  n) 


__  (i+j  +  *•  -  1)  a 
i+j-  (S  +JXT-  *)  a2  (t  +  j  +  1  -  (t  - 

(_t  +  j  +  *  -  1  )  a   (s  +  j)  (1  -  s)  a2 


1  -  1  -  *  +  j  +  1 

+  s-l)«    (a  +  j)  (1  -  s)  a2    (i-s+  !_)  (i 

' 


i-  i-  i- 

and  this  is  the  required  form.  The  relation  between  the  alternate  constituents 
is  obvious  enough,  for  the  substitution  of  j  +  2  for  j  and  n  —  2  for  n  (or  s  —  1 
for  s)  clearly  has  the  effect  of  increasing  each  factor  by  1  in  the  numerators 
and  by  2  in  the  denominators.  As  i  =  k  is  a  fairly  large  number  in  the  direct 
calculation  of  p^'i,  the"  even  constituents  are  small  and  the  calculation  is 
based  on  an  odd  number  of  terms  (generally  five).  With  the  use  of  subtraction 
logarithms  the  process  is  rapid. 

152.  The  next  step  is  to  consider  two  circular  orbits  in  planes  inclined  at 
an  angle  /.  Let  L1}  L2  be  the  longitudes  in  the  two  planes,  reckoned  from 
the  common  node,  and  let 


fji  =  cos       ,     v  =  sn       , 
x  =  Ll  —  Lz,   y  —  LI  +  Lz. 

Then  the  angular  distance  between  the  planets  is  given  by 
cos  H  =  cos  Zj  cos  L2  +  sin  Ll  sin  L2  cos  J 
=  p  cos  x  +  v  cos  y 


11—2 


164  The  Disturbing  Function  [OH.  xiv 

and 

a2  A-1  =  (1  +  of  -  2«  cos  H)  ~  •> 

=  &°>°  +  2  2  ¥>°  cos  ix  +  2  2  6°'  •>  cos  JT/  +  422  &'>  i  cos  we 

e=l  j=l  i=lj=l 

where 

"""    77 

1)  cos  ix  cosjy  dx  dy. 


1    /"""  f77 
=  —  2  I     I 


When  v  is  small  A"1  can  be  expanded  in  powers  of  v.     Thus 

0-2  A"1  =  {1  +  a2  —  2a  cos  #  —  2av  (cos  y  —  cos  #)]  ~  * 


»=o 
or 


where 


It  is  only  necessary  to  compare  the  coefficients  of  f*'«^  in  these  expressions  in 
order  to  have  &•*  as  a  power  series  in  v,  the  coefficients  being  functions  of  a. 
Thus,  for  example,  as  far  as  v2, 

26*-°  =  &*  -    av  6/+1  +  ft*-1  +    a2^2  6'+2  +  4&s'  +  fc-*  -  ... 


It  is  easy  to  continue  these  developments  further,  and  this  is  the  method 
used  by  Le  Verrier  and  Newcomb.  But  its  validity  is  limited.  The  binomial 
expansion  (10)  of  a2A-1  is  convergent  only  when 

1  +  a2  —  2a  cos  x 

2  a  (cos  y  —  cos  x) 

and  since  the  most  unfavourable  case,  cosx=  —  cosy  =  1,  must  be  included 

sin-  J  J  =  v  <  (1  —  a)2/4a. 

It  has  been  proved  by  H.  v.  Zeipel  that  the  same  limit  applies  to  the 
expansion  of  Jacobi's  coefficients  b{'i.  This  condition  is  satisfied  in  all  cases 
by  the  small  inclinations  of  the  orbital  planes  of  the  major  planets. 

153.  Among  the  orbits  of  the  minor  planets,  however,  are  some  whose 
inclinations  to  the  plane  of  Jupiter  exceed  the  above  limit.  It  is  therefore 
desirable  to  find  a  more  general  form  of  development.  Let 

F-*  =  (1  +  a2  -  2ao-)-s  =  2(7/a/l. 


152,  153]  The  Disturbing  Function  •  165 

The  coefficients  Csn  are  polynomials  in  a,  which  are  in  fact  Legendre's  poly- 
nomials when  s  =  ^.     Differentiation  with  respect  to  er  and  log  a  gives 

tfs+2         (1C  n 

V  *     nn  1     i       2 

2sa  ~     da- 

Jfs+2       ,72/7  n 

_,  u/  \JS  _    ,  . 

2-7  o    «"  =  2(s+l)  a 


2sa 

-  2  wC,"an  =  (o-  -  a)  (1  +  a2  -  2acr) 
2sa 

-  2  ?i2(7/an  -  (o-  -  2a)  (1  +  a2  -  2a<r)  +  2  (s  +  1)  a  (a  -  a)2 

2, vet 

=  (o-  +  2sa)  (1  +  a2  -  2a<r)  -  2  (s  +  1)  a  (1  -  a2) 


=  —  s  — 

2s  a 
Hence  (7/'  satisfies  the  differential  equation 

Now  in  the  present  case 

a  =  cos  H  =  fi  cos  x  +  v  cos  y 
and  the  problem  is  to  develop  Csn  in  the  form 

GSH(O-)=  2  AnitjCosixcosjy    -(12) 


where  the  coefficients  A\j,  considered  generally  as  functions  of  /*,  v,  are 
Appell's  hypergeometric  series  in  two  variables  /u.2,  v2.  But  the  solutions 
required  can  be  deduced  from  the  well  known  equation  (11)  by  a  certain 
treatment.  It  will  be  seen  that  this  treatment  is  very  special,  but  it  is 
adequate  for  the  purpose  in  view. 

Let  yu.,  v,  which  are  not  in  fact  independent,  for  /u-  +  v  =  1,  be  considered  as 
functions  of  a  variable  t.  Their  derivatives  with  respect  to  t  will  be  denoted 
by  /jf,  /ji",  v',  v".  Then 

d*C  _  dC  &C 

;r    ~  —  ~~"  LL  COS  &  ~~j      ~r~  ///"  S1H"  OC  ~z     ~ 

ox~  da-  acr- 

cT-C  dC  d*C 

—  -  =  —  v  cos  y  -j-  +  v'2  sin-  y  -j—  - 
9y2  y  do-  y  do-* 

dC  dC 


d*C  x  dC  ,„ 

-^—  =  (/*  cos  «  +  v  cos  y)  -j  —  h  0*  cos  .•»  4-  ^  cos  ?/)2  -^  . 

It  will   now  be  seen  that  if  with    the  help  of  these  equations  a  partial 
differential  equation  can  be  deduced  from  (11),  such  that  <r,  cos#  and  COST/ 


166  The  Disturbing  Function  [CH.  xiv 

do  not  appear  in  it,  a  differential  equation  satisfied  by  A\j  will  be  deducible 
on  comparing  the  coefficients  of  cos  ix  cosjy.     Now 

d*C 
n(n+  2s)  C  =  (yit2  cos2  x  +  v*  cos2  y  —  1  +  2/jt.v  cos  x  cos  y)  -5—^ 

+  (2s  +  1)  (yu  cos  x  +  v  cosy)  j— 

dcr 

=        +     ^  cos2  *  +  "2  cos2  y  ~  l  ~        cos2  *  +  "'2  cos2 


+  -=-    (2s  +  1)  (yii  cos  #  +  i>  cos  ?/)  -  -^  (/*"  cos  a?  +  v"  cos  y) 
$0"  L  IMV 


.  /n 

=  ~i~>  ^2  +  ;n   /*  +  r  -  1  —  —/  (^  -  +  ^  2) 

/*  z;    dr       a2 


, 

,  f  —  ^  )  I  —  >  -^—  --  r  5~7 
oar     fj,  v  oyj 


cos  x  +  j  2sv —  (w"  —  v'*}  \  cos  y 

and  therefore  if 

yLl  V 

2s  4  -  -,  (^  -  l\  =  2.s-  -x  -  — ,  fc-  -  l)  =  JV 

the  equation  takes  the  required  form 

d2C       1   3 


.  .  „ 
w  (TO  +  2s)  C  =  -V-,  1r- 

pv   dP  Vyu,^  o*-2      yu,  v  oy2 

154.     At  present  p,  and  y  are  any  functions  of  t.     Let 

^-  =  (l-pl}(\  -ps),     vz=plp.1. 

Then  it  will  easily  be  found  that  the  first  condition  becomes 
ipfi'vv'M  =  (Pl  -  p2)2  /,//,/  =  0. 

Hence  either  pj  =  p2  or  p2  is  independent  of  t.     The  first  case  has  the  more 
obvious  importance  since  it  gives  directly 

v  =  PJ  =  sin2  £  J,     yu,  =  1  -  p,  =  cos2  1  /. 
The  second  condition  may  be  written 


and  the  right-hand  vanishes  because  //,  +  v  =  1.  Hence  the  method  can  only 
be  pursued  further  when  s  =  £,  but  this  happens  to  be  the  most  important 
special  case.  If  now  t  =  v,  v  =  —  //  =  1,  //,"  =  v"  =  0,  and  the  partial  differential 
equation  (13)  in  C  becomes 


v-~. 
v  oy2  '  dv 


153-iss]  The  Disturbing  Function  167 

On  inserting  the  series  (12)  and  comparing  the  coefficients  of  cosix  cosjy  this 
gives 

n  (n  +  1)  Ani  j  =  -  v  (1  -  i/)  d^' j  +  (^—  +  J-]  A\tj  +  (2i/  -  1)  -^ . 

'  iJ  '  rttt~  \    I     11  i»    /  tj  \  /  y-/i» 


But  the  direct  expansion  of  F  s  shows  that  since  cosix  cosjy  arises  from 
terms  of  the  form  (/u,cos#  +  vcosy)™,  Anitj  must  contain  /u,lV  as  a  factor.  It 
is  therefore  proper  to  write 

4\f«(i-iOMJ?»<j 

and  this  gives,  with  a  little  reduction, 


or 

(^-»>^^  +  ^(*^>1)-«-M:^+W-")(»i+^1+»)^W-^ 

\lll/  \J(jV 

Now  -B\  j  is  a  polynomial  in  v  with  a  constant  term,  and  this  equation  gives 
the  law  of  its  coefficients.  But  the  equation  is  clearly'of  the  form  satisfied 
by  a  hypergeometric  series.  Hence 

Anij  =  c/jLlviF(i+j  —  n,  i+j  +  1+nt  2j '  +  1,  v)  (15) 

where  c  is  a  constant  depending  on  i,j,  n.  This  gives  the  form  of  Hansen's 
development  in  powers  of  a,  namely 

a.2 A"1  =  2    aw .  A\  j  cos  ix  cosjy,     (n  >  i  +  j). 

n,  i,  j 

The  determination  of  the  constant  c  may  be  deferred. 

155.  This  is  the  simplest,  most  obvious  application  of  the  method.  But 
its  possibilities,  though  limited,  are  not  exhausted.  The  first  condition  for 
its  use  is  also  satisfied  by  making  p2  a  constant.  This  may  be  expressed  by 


where  J0  is  to  be  treated  initially  as  constant,  though  finally  it  will  be 
identified  with  ./.  The  relation  /i  +  v  =  1  no  longer  holds  formally,  but  is 
replaced  by 


and  the  result  of  differentiating  this  twice  with  respect  to  t  and  eliminating 
tan  ^ J"0  shows  that  the  right-hand  side  of  the  second  condition  (14)  is  1. 
Therefore  s  =  1.  At  first  sight  this  case  has  no  present  interest,  since  s  is 
not  half  an  odd  integer,  but  the  reason  for  considering  it  further  will  be 
seen  later. 

The  development  will  be  in  powers  of  sin2|J  as  before,  but  it  will  be 
convenient  first  to  make  t  =  ^J,  so  that 

//  =  —  sin ^J cos  £  J0,     v  =  cos | ./sin  £  J0,     /j,"  =  — 


v   —  —  v. 


168  The  Disturbing  Function  [on.  xiv 

Then  the  partial  differential  equation  (13)  for  C  becomes 

n  (n  +  2)  G  =  -  '       -  sec2 1-^-9-  cosec2 1  --  -  2  cot  '2t  •=-  . 
ot2  das2  dy2  dt 

The  form  of  the  solution  resembles  the  previous  case,  suggesting 

(7=2  iiiv^Tnitj  cos  ix  cosjy 

and  the  comparison  of  coefficients  of  cos  ix  cosjy  after  the  substitution  gives 


Now  let  the  independent  variable  be  changed  to  r  =  sin2£  =  sin2  \  J,  so  that 


-  ,        2 

dt  dr       dt2 

and  the  previous  equation  becomes 


4  (T2  ~  T)        ~  +  4  {(t+  j  +  2)r-  O'+l))          J  +  (t+j  -n)(i+j+  2  +  n)T\}  =  0. 


Now  T\j  is  a  polynomial  in  T  with  a  constant  term,  and  this  equation 
determines  the  formation  of  its  coefficients.  But  again  it  is  an  equation 
of  the  type  satisfied  by  a  hypergeometric  series.  Hence 


where  ct  is  independent  of  r.  But  //,  and  v,  and  therefore  T\j,  involve  J9 
symmetrically  with  /,  and  therefore  it  is  evident  that  Cj  contains  as  a  factor 
the  same  polynomial  with  T  replaced  by  TO  =  sin2  1  Jo.  Hence 

T\j  =  c2F(r0)F(r) 

where  c2  is  a  constant  independent  of  T  and  TO.  This  is  clearly  general, 
whatever  the  values  of  J  and  ,/0.  A  return  to  the  actual  problem  can  now 
be  made  by  putting  JU=J,  and  then  r  =  v  and 

+  n     ... 

'  ^ 


2       '  2 

which  gives  the  form  of  expansion 

«22  A~2  =  2  an  .  Tnu  /*V  cos  ix  cosjy 

n,  i,  j 

(i+j<  n).     The  form  of  proof  is  essentially  that  of  Stieltjes.     The  squared 
(terminating)  hypergeometric  series  is  a  polynomial  of  Tisserand. 

The  more  general  utility  of  this  result  will  now  be  easily  seen.     For 
a22A-2  =  (1  +  a2  -  2«  cosH)~l  =  (1  -  OLZ}~I  (1  -  az'1}^ 
=  {z  (1  -  az)->  -  z~l  (1  -  a*-1)"1}  (z  -  z~1}-1 
=  2  an  (zn+l  -  z~n~1}  (z  -  z~l)~l 


155,  ise]  The  Disturbing  Function  169 

Hence,  by  comparing  the  coefficients  of  a", 

sin  (n  +  1)  H/sin  H  =  2  Tnij/j.ivj  cos  ix  cosjy. 
But 

CO 

(a.2~l  A)~s  =  $bg°  +  2  &/*  cos  nH 
i 

=  ^6g°  +  2  %bgn  {sin  (n  +  1)  H  —  sin  (n  -  1)  Ity/sin  7/ 
and  therefore 

(a2~]  A)-*  =  li;1  +  12  b,n  2  (T?_.  -  r;:2)  /*V  cos  iar  cosjy     . .  .(16) 

which  is  Tisserand's  development  in  a  series  of  Laplace's  coefficients. 

156.     To  complete  the  result  it  is  necessary  to  find  the  numerical  factor  c2. 
Now  the  final  term  of  F(—oi,  /3,  7,  x},  a.,  /3,  7  being  positive  integers,  is 


Hence  the  term  containing  the  highest  power  of  v  in  T\jfj,ivj  is 


2 

V*. 


But 

a22  A~2  =  {1  +  a2  —  2a  cos  x  —  2av  (cos  y  —  cos  x)}~1 

=  2  (2ay)m  (cos  y  -  cos  #)m  (1  +  a2  —  2a  cos  a?)-™-1 
and  the  highest  power  of  v  associated  with  an  is  given  by  the  terms 
(cos  y  - 


=  2 


when 


The  same  terms  appear  in  the  form 

2  Tn;  j  fj,{vJ  cos  we  cos  jy  =  K^  T\  }  fjfvJ^rf 


where  K  =  1  when  i  and  j  =  0,  K  =  1  when  i  or  j  =  0,  and  K  =  |  otherwise.  The 
highest  power  of  v  has  already  been  found  in  this  form,  and  comparison  of  the 
coefficients  of  vn%'rf  gives  finally 


The  development  (16)  is  now  completely  defined. 


170  The  Disturbing  Function  [CH.  xiv 

The    numerical   factor   c   in    Hansen's   development  (15)  can  be  found 
similarly.     For  the  term  containing  the  highest  power  of  v  in  A\  j  is 


On  the  other  hand  the  terms  associated  with  an  and  the  highest  power  of  v 
in  a2A-1  are  by  (10)  contained  in 


and  these  are  now  known.     As  before,  the  coefficients  of  i/nf'V  in  the  two 
forms  of  aaA""1  can  be  compared,  and  thus 

(2n)  !  <2y)  !  (-l)'(n  !)2  F  («  +  J) 


(n  +  j  -  i)  !  (n  +  t'  +  j)\      H  {ft  (n  ±  i  ±.7)]  I}'  T  (n  +  1) 

where  II  denotes  the  product  of  four  factorial  factors.     Now  ^(n  —  i—j)  is 
an  integer,  n  —  i—j  is  even,  and.  the  sign  is  the  same  on  both  sides.     Also 

r(n  +  l)  =  n\,     22WF  (n  +  i)  .  n  I  =  T  (£)  .  (2n)  !. 
Hence  finally 


_ 


(2?)  !    [ 

j 

which  completes  the  determination  of  Hansen's  development. 

The  results  obtained  for  inclined  circular  orbits  may  now  be  summarized. 
Since 

cos  ix  cosjy  =  cos  i  (L^  —  L2)  cosj  (Li  4-  L2} 

=  |  cos  [(i  +  j)  LI  -  (i  -j)  L2]  +  |  cos  [(i  -j)  Ll  -  (i  +j)  La] 

it  is  possible  to  write 


where  log\1  =  t//J,  logX2=tZ2;  and  it  has  been  shown  how  the  coefficient 
A(pl,p2)  can  be  developed  (1)  in  powers  of  j'  =  sin2^J,  (2)  in  powers  of 
a  =  Oi/a2>  (3)  as  a  series  in  Laplace's  coefficients. 

157.  The  preceding  developments  of  A"1  or  A~2S  apply  to  circular  orbits, 
but  they  are  not  on  that  account  to  be  regarded  as  mere  approximations  to 
the  forms  actually  appropriate  to  the  orbits  of  the  solar  system.  On  the 
contrary  they  constitute  the  essential  source  from  which  the  latter  forms 
must  be  generated  by  the  most  convenient  means.  Now  quite  generally 

A2  =  7-j2  +  r22  —  2rjr2  cos  H 

and  LI,  L2  must  be  replaced  by  (at+Wi,  w.2  +  w.2,  where  tol}  w.>  are  the 
longitudes  of  perihelion  reckoned  from  the  common  node,  and  wlt  w2  are  the 
true  anomalies.  When  the  eccentricities  e},  e.,  vanish  the  radii  ?•,  ,  r.,  become 


156,  is?]  The  Disturbing  Function  17fl 

the  mean  distances  a3,  a2,  and  wl,  w.2  can  be  identified  with  the  mean 
anomalies  M1}  M2.  The  corresponding  value  of  A  may  be  written  A0. 

Taylor's  theorem  can  be  expressed  in  the  familiar  symbolical  form 
f(x  +  y)  =  exp.  (y  3-)/(*)  =  exp.  (yD}f(x} 

\       U,t  / 

which  means  simply  that  if  the  exponential  function  be  exparicfccfas  though  yD 
were  an  algebraic  quantity,  the  result  otherwise  known  to  be  true  is  formally 
reproduced.  Thus  generally, 

/(«!  +  ylt  ara  +  2/2,  ...)  =  exp.  (ylD1  +  y.2D2  +  ...)/Oi,  a?2,  •••) 
where  Dr  operates  on  xr  alone.     Now  when  el  =  e2  =  0, 

Ao"1  =/(«!,    02,   A,    A) 

is  an  expansion  of  which  the  form  has  been  completely  determined.  The 
more  convenient  developments  refer  not  to  r  —  a  but  r/a,  and  the  change 
from  the  argument  a  to  the  argument  r  is  made  additive  by  taking  log  a  as 
the  variable  instead  of  a.  Thus  in  the  present  case 

x1  =  \oga1,         #2  =  log  a,,         xs=-L1  =  to-i  +  M1,    x4  =  Lz=  w2  +  M2 
y1  =  logr1/o1,     ya=logra/as,     y3  =  w1-Ml,.  y4  =  w2-M2 

n          a          ,   8       n          8  8 

-L'l  —  ^\  i  —  ^l-i         >        -^2  —  -~,  i  —  ^2  o  -  j 

9  log  aj         dc^  9  log  a2         9a2 

jr),  =  A  =  tX  A          !»'•-  8   ,.  A.— 

9//!  9Xj  '  9//2  "  9^2 

Then  generally 


=  exp.  [log  ~  -  A  +  log  J  .  A  +  (w,  -  3fO  D, 

(/,  a-2 

But  in  the  notation  of  Hansen's  coefficients  (§  45) 


where  log  a?  =  tw,  log  0  =  iM.     Hence  in  a  corresponding  symbolic  notation, 
since  log  ac/z  =  i(w  —  M), 

A-i  —  V   YD>  '~ll>3     i          VD-'~lI>i 


Simplifications  are  now  possible  owing  to  the  form  of  /.     In  the  first 
place  A0-1  is  homogeneous,  and  of  degree  —  1,  in  al}  a2.     Hence 


A  +  D2  =  «!  —  +  a.2 


^- 
da., 


172  The  Disturbing  Function  [OH.  xiv 

But  further /has  been  expanded  in  the  form 

and 

so  that  A>  A  can  be  replaced  by  ip1}  ip.2,  and  A.  A  do  not  operate  on  Xa,  X2. 
Hence  the.  symbolic  form  of  the  complete  expansion  becomes 

Pi ,  P*  i,  j 

where   log  Xa  =  i  (w1  +  M^,   log  Xo  =  i  (&>2  +  M2),   \ogz1  =  iM1,\ogz.2  =  iM2,  and 
the  symbols  X  are  respectively  functions  of  e, ,  A  and  e2,  A- 

158.  This  leads  immediately  to  Newcomb's  operators  as  defined  by 
Poincare.  For  the  functions  X  can  be  expanded  in  positive  powers  of  e, 
so  that 

where  m1  -    i  ,m2-  \j  |  =  0,  2,  ...,  since  Xin>m  is  of  the  order  fe1*-™  at  least. 
The  operators  II  are  combined  by  Newcomb  in  the  notation 

but  the  combined  symbols,  though  tabulated  by  him  over  a  wide  range,  seem 
to  present  no  practical  advantage  over  the  constituent  operators. 

The  final  form  of  the  development  of  A"1  can  therefore  be  written 


and  the  completion  of  this  part  of  the  problem  depends  on  the  practical 
treatment  of  Newcomb's  operators  II,  which  are  polynomials  in  D,  p  of 
degree  in,  with  numerical  coefficients. 

The  definition  of  the  symbols  is  given  by 

2  n,-  (A  p)  e»* 


Hence  in  particular 

2  n^  (D,  0)  em^  =  (-)  ,     2  n/»  (0,  ^)  em 

»»,  i  \O>/          m,  i 

and  therefore  , 

2  n^  (D,  p)  emzi  =  2  H,-"1  (D,  0)  ew^  .  2  H  /  (0,  p)  e»^'. 

«l,  i  m,  i  ii.  j 

Comparison  of  the  coefficients  of  emzi  on  both  sides  then  gives 

n/»  (A  p)  =  2  n/  (A  o>  n;qw  (o,  P) 
«,  j 

where  w  =  0,  1,  ...,  m,  and  j  has  all  the  values   which   make   n  —  \j     and 
»?  —  n  —  1  1  —  j   positive  integers  (including  0).     This  formula,  due  in  another 


157-159]  The  Disturbing  Function  173 

notation   to   Cowell,  makes  the  calculation   of  IT/'1  (D,  p)   depend   on    the 
expansion  of  r/a  and  xv. 

But  these  are  known  forms.     The  first  is  given  by  (22)  in  Chapter  IV. 
Means  of  deriving  the  latter  have  been  given  in  §  45.     In  fact 


and  therefore  it  is  necessary  to  expand  X.'^*  in  powers  of  e  and  the  resulting 
coefficients  will  represent  11^(0,  p).  They  are  purely  numerical  and  can  be 
tabulated  for  all  moderate  values  of  m,  i  and  p.  Other  methods  have  been 
suggested  to  facilitate  the  calculation  of  Newcomb's  operators.  But  the 
above  will  suffice  to  make  clear  the  principles  involved. 

159.     The  disturbing  function  due  to  the  complete  action  of  a  single 
planet  can  now  be  considered.     By  (3)  of  §  23  this  is 

R  =  Gm  \  -r-  -  — .  (xx  +  yy'  +  zzr 

\  A        T  ' 

where  (x,  y,  z),  (x,  y',  /)  are  the  heliocentric  coordinates  of  the  disturbed  and 
disturbing  planets ;  r'  is  the  radius  vector  of  the  latter.  The  constant  G 
may  be  reduced  to  unity  by  the  choice  of  appropriate  units,  and  the  dis- 
turbing mass  m'  may  be  understood  as  a  common  factor  to  be  restored 
ultimately.  Thus 

R  =  (,-2  +  r's  _  2rr'  cos  H)  ~  *  -  rr'~*  cos  H 

where  H  has  its  previous  meaning,  the  mutual  elongation  of  the  two  planets 
as  seen  from  the  Sun.  The  principal  part,  already  discussed,  is  symmetrical 
in  r,  r',  but  the  indirect  part  is  not  so.  Hence  a  distinction  must  be  drawn, 
according  as  the  disturbing  planet  is  superior,  when  r  =  r1;  r'  =  r2,  or  the 
disturbing  planet  is  inferior,  when  r  =  r.2}  r' =  rl.  Now  when  the  eccen- 
tricities vanish,  by  §  152, 

(/.A-1  =  6°'°  +  2bl>n  cos  x  +  260'1  cos  y  +  ... 

cos  H  =          \      /*  cos  x  +       v  cos  y 

and 

R  —  A"1  =  BR  =  —  aa'~2  (//.  cos  x  +  v  cos  y) 

is  the  correction  required  to  change  A"1  into  R.  This  can  be  effected  by 
giving  corrections  to  b1'0  and  60-1,  thus 


=  —  a  (a'  >  a) ;  —  or-  (a  >  a') 

where  a  <  1  always  and  a'  is  the  mean  distance  of  the  disturbing  planet.     If 
these  corrections  are  carried  into  the  expansion  in  terms  of  v  (§  152),  as  used  in 


174  The  Disturbing  Function  [CH.  xiv 

the  chief  planetary  theories,  it  will  affect  the  Laplace's  coefficients  only  to 
this  extent  : 

8b£  =  -  a,        &b%°  =  -2          (a'  >  a) 

$b£  =  -a-2,     S64°  =  -2a-3     (a  >  a') 

for  it  is  easily  verified  that  these  changes  will  give  the  required  corrections 
to  61'0,  &M.  In  the  exponential  form  they  apply  equally  to  b~^°,  b°>~1, 
and  b,~\  Thus  the  indirect  term  is  very  simply  incorporated  in  RQ,  in 

which  e1=e2  =  0,  and  the  full  expansion  of  R  in  terms  of  the  eccentricities 
can  then  be  deduced  in  the  manner  explained  for  the  development  of  A 
from  A0. 

It  is  most  important  to  remark  that  while  the  indirect  part  modifies  the 
coefficients  of  certain  elementary  periodic  terms,  it  affects  in  no  way  the 
constant  term  which  is  independent  of  the  time. 

160.  Another  order  of  development  is  possible  by  expanding  A"1  initially 
in  terms  of  rj/r2.  If  this  ratio  is  small,  as  in  the  case  of  the  solar  perturba- 
tions of  the  lunar  orbit,  this  method  has  great  advantages.  By  §  153  this 
expansion  takes  the  form 

A"1  =    2   r^  r2-n~1  A  \  j  cos  ix  cos  j  y 

n,  i,  j 

where  A\j  is  given  by  (15)  and  x,  y  have  their  true  meanings, 
Wj  +  W2  =  co,  +  Wi  +  (&>2  +  w2). 

It  is  more  convenient  to  use  the  exponential  form,  and  with  a  slight  change 
of  notation  for  the  coefficients, 

A-1  =      2      r^r^-^An  (Pl,  p,)  tf*pf* 

n,  pt  ,  p2 

where  log  /^  =  i(Wl  +  w^,  log  /i.2  =  i  (w.2  +  w2),  \p1-p,\  =  2i,  p1  +  p2  =  2j 
and  n  —  \p1  ,  n  —  \p2  \  are  even  positive  integers.  Hence 


n,  p,  ,  p, 

where  logXj^  *(»!  +  MJ,  logX2=t(<M2+iT2),  \o^zl=iMl,  \ogZs=iM2, 

log  x2  =  tWj.     But  this  form  can  clearly  be  expressed  in  terms  of  Hansen's 

coefficients.     Thus 

A-*=    S      2  0^0.^4.^,  i^y^V'^4ziXl'^*Vl 

n>  Pi  )  Ps  9\>  </2 

where  qlt  qz  have  all  integral  values,  positive  and  negative,  and  the  symbols  X 
are  respectively  functions  of  e^  ,  e2,  while  An  (pli  p2)  is  a  function  of  v  =  sin2  \J 
which  has  been  determined. 

The  indirect  part  of  the  disturbing  function,  when  r,  (<  r2)  refers  to  the 
disturbed  body,  is  clearly  allowed  for  by  simply  excluding  the  terms  cor- 
responding to  n  =  1,  for  these  are  equal  to  ?-1?-3~2cos  H. 


loo-lei  ]  The  Disturbing  Function  175 

By  either  method  the  fundamental  importance  of  Hansen's  coefficients 
and  their  relation  to  Newcomb's  symbolic  operators  is  clearly  seen.  Numerical 
developments  of  their  coefficients  according  to  powers  of  e  have  been  calculated 
by  several  authors,  including  Cayley,  Newcomb  and,  for  the  purposes  of  the 
lunar  theory,  Delaunay. 

161.     It  has  been  seen  that  the  generating  expansion  is  of  the  form 
R  =  'Z  2A/uiVv'i  cos  px  cos  qy 


where  L  =  &>  +  M,  L'  —  w  +  M'.  The  subsequent  process  introduces  e,  e  into 
the  coefficient  A,  which  already  contains  powers  of  v  —  sirf^J,  and  adds 
multiples  of  M,  M'  to  the  argument.  In  the  ordinary  notation  for  the 

elements, 

w  =  OT  —  I)  •  —  ;Y,      to'  =  tzr'  —  II*  —  % 

where  ^,  %'  are  the  distances  of  the  intersection  of  the  orbits  from  their 
ecliptic  nodes.  Hence  R  takes  the  form 

R  =  2  AftPtf  cos  \hM  +  h'M'  +  (p  +  q)  O  -  fl) 

-(p-q)  (V>  -  ft')  -  p  (X  -  %')  -q(X+  x')]. 

Now  the  two  orbits  with  the  ecliptic  form  a  spherical  triangle  ABC  in  which 
«  =  X'>     b  =X>  c  =  ns  -  flj 

^=1,  B  =  7T-i',        C  =  J 

where  i,  i'  are  the  inclinations  of  the  orbits  to  the  ecliptic.  Hence,  as  in  §  67, 
if  the  intersection  be  taken  as  the  ascending  node  of  the  disturbing  orbit  on 
the  disturbed  orbit, 

sin  $(x  +  %')  sin  £  J=  sin  £  (H'  -  II)  sin  £  (i'  +  i) 
cos  ^  (%  +  %')  sin  ^  /=  cos  i  (II'  —  O)  sin  ^  (i'  —  i) 
sin  ^  (%  -  %')  cos  |  J  =  sin  -|  (II'  —  H)  cos  £  (*'  +  1) 

cos  ^  (%  —  %')  cos  ^  J=  cos  |  (fl'  —  O)  cos  \  (i!  —  i) 
and  therefore 

v  exp.  ^^  (%  +  %')=sin  \i!  cos  |i  exp.  ^  t  (ft'—  H)—  sin  ^i  cos  \i'  exp.—  \i  (II'—  H) 
/i^exp.  ^t(%  —  %')=cos^i'cos^-iexp.  |t(H'  —  O)+sin^ism^''exp.—  ^i(H'  —  II). 
It  follows  that 

1/9  cos  ^  (%  +  %')  =  2  6g  cos  s  (H'  —  H),      yi  sin  ?  (%  +  %')  —  ^  ^«  gin  s  (&'  —  ^) 
/i^  cos  p  (%  -  %')  =  2  ag  cos  s  (H'  —  H),    ^  sin  p  (^  —  ^')  =  2  as  sin  s  (H'  —  O) 

where  as,  bs  represent  simple  coefficients  involving  i,  i'.  Thus  %  ±  %'  can  be 
eliminated  from  R,  which  now  takes  the  form 


R  =  S  A  cos  [hM  +  h'M'+(p  +  q)  (vr-fl)-(p-  q)  («r'  -n')-(s  +  s')  (IT  -  fl)] 


176  The  Disturbing  Function  [CH.  xiv 

where  A  now  contains  a,  a,  e,  e',  i,  i'  and  also  powers  of  v.  But  from  the 
above  analogies  of  Delambre, 

v  =  sin2  £  (IT  -  n)  sin2  £  (i'  +  i)  +  cos-  £  (IT  -  H)  sin2  i  (*'  -  i) 
=  ^  (1  —  cos  t  cos  i)  —  ^  sin  i  sin  i'  cos  (fi'  —  £1). 

Hence  these  powers  of  v  can  be  removed  from  the  coefficient  without  altering 
the  form  of  the  arguments,  which  are  only  changed  by  the  addition  of  some 
multiples  of  O'  -  H.  Thus  finally 

R  =  2  A  cos  [hM  +  h'M'  +gv  +  #V  +/O  +f&] 

=  2  A  cos  [h  (nt  +  e)  +  //  (V*  +  e')  +  </BT  +  g'vr'  +/&  +f'&] 

where  the  coefficient  A  is  now  a  function  of  a,  a,  e,  e',  i,  i  only,  and  the 
argument  contains  the  six  elements  H,  ft',  CT,  a/,  e,  e  and  the  time.  And 
this  is  the  final  form  of  the  disturbing  function,  involving  the  twelve 
elements  of  the  two  orbits  explicitly,  and  expressed  in  the  desired  way. 


CHAPTER   XV 

ABSOLUTE     PERTURBATIONS 

162.  The  disturbance  of  a  purely  elliptic  motion  may  be  illustrated  in 
a  quite  elementary  way  by  supposing  the  motion  to  take  place  in  a  resisting 
medium.  Let  the  tangential  resistance  per  unit  mass  be  av/r'2,  where  v  is  the 
velocity  and  r  the  radius  vector,  so  that  the  radial  and  tangential  components 
are 

av   1  dr         a.  dr  av  r  dd         a.  dd 

"/*^      ?J     Ci  /  7*^    (1  /  7"^      7J     fl  /  V     ft  1" 

When  other  powers  of  v  and  r  are  assumed  in  the  expression  for  the  resistance 
the  general  results  are  very  much  the  same,  and  this  simple  form  is  sufficiently 
typical  to  represent  fairly  an  interesting  problem. 

Let  u  be  the  reciprocal  of  r  and  8  W  the  work  done  by  external  forces  in 
a  small  radial  or  transversal  displacement.  Then 

du          dW  dd 


-  u-  -5—  =  -fjuu         ~]-,         ^-Q  -r- 

ou  dt  08  at 

where  ju,  is  the  constant  of  attraction ;  and  the  kinetic  energy  is  T,  where 


Hence  the  equations  of  motion  are 

d  A  du 

•j-  (u    u)  4-  2>u    u  -\-u    v2  =  fj,  —  au~ 2  -5- 
dt  dt 

"Vw  -,M 

j     \(ju        U  i  —  C*      j      . 

at  dt 

Now  let 

u~*d  =  H,     -t  =  Hu*^ 

and  the  first  equation  of  motion  becomes 

u  i  d  f  TT     *du\  .  ^TTn     ,  fduV  .   „.„  rrdu 

Hu1  -JQ  (  H u~2 

or 


P.  D.  A.  12 


178  Absolute  Perturbations  [OH.  xv 

But  by  the  second  equation  of  motion 

H=h-a0 
where  h  is  constant.     Hence 

dzu  p       _0 

d0*  +        (h-a0?~ 

It  is  enough  to  retain  the  first  power  of  a,  so  that 


and  the  integral  is 

u  =  ph-2{l+ecos(0-y)  +  2cth-10}  .....................  (1) 

where  e  and  7  are  constants. 

163.  The  osculating  ellipse  at  the  point  6  =  01  is  obtained  by  supposing 
the  resisting  medium  to  disappear  at  this  point  and  the  subsequent  motion 
under  the  central  attraction  to  be  undisturbed.  The  path  is  then 

u  =  pr1  {1  +  &i  cos  (6  -  70}- 

The  motion  at  the  instant  is  the  same  in  the  actual  trajectory  (1)  and  in  this 
ellipse,  and  thus  0  =  01}  u  =  Ui,  u  and  0,  and  therefore  H  =  H^  and  dujdd  are 
the  same  for  both  curves.  Let  /u/r~2  =  p~l.  Now  Hl  is  the  constant  of  areal 
velocity  in  the  ellipse,  and  hence 


prL 

To  the  first  order  in  a.  then 

j 

Again,  by  equating  the  values  of  u  and  dujdd, 

pr1  {1  +  el  cos  (0j  -  70]  =  p-1  [l  +  e  cos  (6l  -  7)  +  2a^~1  6^ 
Pi~l  {   —elsin(01  —  jl)}=p~l{    —  e  sin  (6^  —  7)  +  2a/z~1} 
and  to  the  first  order  in  a 

«!  cos  (0j  -  70  =  e  cos  (0,  -  7)  -  2aA~1e01  cos  (0!  -  7) 

ej  sin  (#j  —  70  =  e  sin  (^  —  7)  —  2a/i~1  —  2a/t"1e01  sin  (0j  —  7). 
Hence 

0!  cos  (7!  —  7)  =  e  —  2ah~le01  —  2ah~l  sin  (0!  —  7) 

ex  sin  (71  —  7)  =  2ah~l  e  cos  (#1  —  7) 
and,  still  to  the  first  order, 

[e6l  +  sin  (#1  —  7)} 
1  cos  (0j  —  7). 

Between  these  terms  an  important  practical  distinction  is  at  once  apparent. 
That  in  A^  depending  on  0^  will  diminish  the  eccentricity  indefinitely  until 
the  orbit  becomes  circular.  It  is  a  secular  term.  The  other  terms  are 


162-164]  Absolute  Perturbations  179 

periodic,  and  when  a  is  small  their  effect,  not  being  cumulative,  is  small  also. 
In  practical  applications,  to  Encke's  comet  for  example,  they  can  be  neglected. 
Then  A7j  =  0  and  the  direction  of  the  apsidal  line  is  unaffected  by  the  resist- 
ing medium. 

In  a  complete  revolution  the  secular  effects  are  given  by 


ei        Pi 

and  the  corresponding  changes  in  the  mean  motion  and  the  mean  distance  are 
given  by 

A/>!  __  3A«!_     3  A/)!     3el&e1      \  +  e^  6-rra 
%  =  ~2~aT=  ~2"^7  ~  1  -  e?  ~  1  -  ef  '  ~h 

since  a1=p1  (1  —  e*)~l.  Thus  the  most  important  effects  of  a  resisting  medium 
are  a  steady  increase  in  the  mean  motion  and  a  steady  decrease  in  the  mean 
distance,  which  must  ultimately  bring  the  disturbed  body  into  contact  with 
the  centre  of  attraction. 

164.  This  simple  example  has  been  chosen,  apart  from  its  intrinsic 
interest,  because  it  illustrates  certain  important  points.  There  is,  in  the  first 
place,  the  osculating  or  instantaneous  ellipse,  which  is 

p^u  =  l  +  e1  cos  (6  —  7j) 
and  not 

pu  =  1  +  e  cos  (6  —  7). 

The  latter  is  a  definite  curve  which  may  be  called  an  intermediate  orbit  and 
may  serve  usefully  as  a  curve  of  reference.  Indeed  it'  has  been  so  used  in 
what  precedes.  But  it  is  not  the  osculating  orbit  at  any  time.  There  is  also 
the  distinction  drawn  between  periodic  and  secular  disturbances  in  the  motion, 
of  which  the  former  may  be  relatively  unimportant  compared  with  the  latter 
because  these,  however  slow,  are  cumulative  in  effect. 

The  general  nature  of  disturbed  planetary  motion  can  now  be  considered. 
For  two  planets  only,  the  disturbing  function  has  the  form,  found  in  the  last 
chapter, 

R  =  2F(a,  a,  e,  e,  i,  i')  cos  T, 

T=  [h  (nt  +  e)  +  h'  (n't  +  e')  +  g<&  +  g'ts'  +/Q  +  /'ft'] 

where  (a,  n,  e,  i,  ft,  BT,  e)  are  the  elements  of  the  disturbed  orbit,  (a,  ri,  e,  i', 
fl',  CT',  e')  the  elements  of  the  disturbing  orbit.  The  equations  of  §  139  are 
now  available  for  finding  the  variations  of  the  elements.  In  accordance  with 
the  artifice  explained  in  §  140  the  mean  longitude  e  is  taken  in  a  special 
sense  there  defined,  and  a  in  the  coefficient  and  n  in  the  argument  of  any  term 
are  treated  as  independent  in  forming  the  partial  differential  coefficients  of  R. 
Therefore 

dR     dR     dR 

da  '    de  '     di 

12—2 


180  Absolute  Perturbations  [CH.  xv 

are  all  of  the  form  2(7  cos  T,  and 


are  all  of  the  form  2(7  sin  T,  where  T  is  the  argument  of  the  term.     Hence 
the  equations  for  the  variations  are  themselves  of  the  form 

?£-24nnr,.. 

at 


In  the  first  approximation  the  right-hand  members  (which  contain  the  dis- 
turbing mass  as  a  factor)  are  calculated  with  the  osculating  elements  of  both 
orbits  for  a  certain  epoch,  and  these  elements  are  treated  as  constant.  The 
equations  can  then  be  integrated,  and  in  fact 

8,  a  =  -  2  C,  cos  Tj(hn  +  h'ri),  .  .  . 


These  are  the  absolute  perturbations  of  the  first  order.  Similarly  the  pertur- 
bations of  the  first  order  in  the  masses  can  be  calculated  for  all  the  disturbing 
planets  concerned  and  the  results  can  be  combined  by  addition. 

165.  Each  term  in  the  perturbations  represents  a  distinct  inequality  in 
the  motion  of  the  disturbed  planet.  It  will  now  be  seen  that  the  inequalities 
are  of  two  kinds.  The  multipliers  A,  A'  have  all  integral  values,  positive  and 
negative,  including  0.  When  A  =  A'  =  0  the  disturbing  function  R  is  reduced 
to  that  part  which  does  not  contain  the  time.  Thus 

da  d£l 


and  the  inequalities  are  secular.  From  the  present  limited  point  of  view  they 
will  increase  indefinitely  and  in  the  course  of  time  will  modify  the  conditions 
of  the  planetary  system  profoundly,  uncompensated  by  any  check. 

But  one  remark  can  be  made  immediately.  The  most  important  element 
as  regards  the  stability  of  the  system  is  clearly  the  mean  distance  a.  Now 
when  A  =  A'  =  0,  not  only  does  t  disappear  from  R  but  also  e.  Hence 

<fa=aR 

dt  ~  8e 

and  in  the  previous  set  of  equations  Cl  =  0.  There  is  therefore  no  secular 
inequality  in  a  of  the  first  order  in  the  masses.  How  far  this  important 
theorem  can  be  extended  to  the  higher  orders  must  be  seen  later.  It  follows 
that  the  mean  motion  n  is  also  free  from  any  secular  inequality  of  the  first 
order. 


164-166]  Absolute  Perturbations  181 

The  other  inequalities,  when  h  and  h'  are  not  both  zero,  are  evidently 
purely  periodic,  unless  hn  +  h'n  =  0.  The  meaning  of  this  qualification  is  that 
the  mean  motions  must  not  be  commensurable.  Now  mean  motions  are  never 
commensurable,  except  perhaps  instantaneously,  since  in  fact  they  are  not 
constant.  But  there  are,  as  it  were,  degrees  of  incommensurability.  In  any 
case  integers  can  be  found  to  make  hn  +  h'n  smaller  than  any  assignable 
quantity.  If  the  incommensurability  of  n,  n  is  high,  tha  corresponding 
integers  h,  h'  will  be  large.  In  general  the  coefficients  in  R  which  correspond 
to  arguments  of  a  high  order  diminish  rapidly  with  the  order.  Then  the 
occurrence  of  a  small  divisor  hn  +  h'ri  on  integration  will  have  no  very  serious 
effect.  But  if  the  incommensurability  of  the  mean  motions  is  low,  this 
divisor  may  become  very  small  for  quite  moderate  values  of  h,  h',  and  a  fairly 
small  term  in  the  disturbing  function  may  be  greatly  magnified  by  integration. 

Thus  in  the  case  of  Jupiter  and  Saturn 

5n  -  2ri  =  Ti/30  =  n'/74 

nearly,  and  this  fact  causes  a  considerable  inequality  in  the  motion  of  both 
planets,  with  a  period  of  nearly  900  years.  The  period  of  such  an  inequality 
is  27r/(/m  +  h'n')  and  therefore  inequalities  of  the  class  just  considered  are 
always  connected  with  long  periods.  They  hold  an  intermediate  place  between 
ordinary  periodic  inequalities  and  secular  inequalities. 

The  mean  longitude  is  affected  in  a  double  degree.     For  (§  140)  this  is 

6-f-    ndt  =  e  +  p 

where 

d*p  3  d-R  ^  n  .  T 
j^  =  -—2  ^-  =  SOsmT 
at2  a2  3e 

and  therefore 

S,p  =  -  2  G  sin  Tj(hn  +  h'nj. 

The  long-period  inequalities  in  the  other  elements  have  the  divisor  hn  +  h'n' 
in  the  first  degree  only.  Hence  the  principal  effect  is  to  be  observed  in  the 
mean  longitude. 

166.  It  is  in  the  next  place  necessary  to  consider  the  perturbations  of 
the  second  order  in  the  masses,  for  the  first  approximation  does  not  in  general 
suffice,  and  in  the  theories  of  Jupiter  and  Saturn  it  is  even  necessary  to  go 
beyond  the  third  order.  It  is  convenient  to  write 


a  =  a0 


where  a0,  .  .  .  ,  e0,  a0',  .  .  .  ,  e0'  are  the  osculating  elements  for  a  chosen  epoch,  and  S1 
indicates  the  perturbations  of  the  first  order,  the  derivation  of  which  has  been 


182  Absolute.  Perturbations  [CH.  xv 

explained,  &2  those  of  the  second  order,  and  so  on.     The  equations  for  the 
variations  of  the  elements  can  be  written,  for  example,  in  the  form 


,  ,    ,. 

-TT-  =  -  ^-.  —  .  -^  =  m'f(a,  a  ,  .  .  .  ,  p  +  e,  p  +  e  ) 
at      cos  </>  sm  t    9i 

and  after  substituting  the  above  expressions  for  a,...,  e   and  expanding  by 
Taylor's  theorem, 


The  reduction  of  the  right-hand  side  to  a  suitable  form  will  be  readily 
understood  in  general  terms,  apart  from  the  complexities  which  will  naturally 
arise  in  the  practical  calculation,  and  a  simple  integration,  requiring  the 
introduction  of  no  arbitrary  constant,  will  give  the  expression  of  S2  ft.  Similarly 
the  perturbations  of  higher  orders,  so  far  as  they  are  of  sensible  magnitude, 
can  be  found  successively,  when  those  of  the  lower  orders  have  been  deter- 
mined, for  all  the  elements. 

167.  The  general  form  of  the  results  will  now  be  apparent.  In  the 
first  order  the  inequalities  are  of  the  forms 

A  cos  (vt  +  h),     At 

only.  In  the  higher  orders  the  terms  obtained  by  the  algebraic  composition 
and  subsequent  integration  of  these  two  forms  will  clearly  belong  to  one  of 
the  three  types 

A  cos  (vt  +  h),     A  ttn,     A  tm  cos  (vt  +  h) 

which  may  be  called  respectively  periodic,  purely  secular  and  mixed  terms. 
The  term  order  may  be  retained  to  denote  the  degree  a  of  A  in  the  masses. 
As  A  is  also  a  function  of  the  eccentricities  and  inclinations,  which  are  also 
in  general  small  parameters,  it  may  be  limited  to  a  homogeneous  function  in 
these  parameters.  Then  the  degree  of  the  term  is  the  degree  of  this  function 
and  represents  its  order  in  respect  to  the  eccentricities  and  inclinations. 

A  further  classification  is  used  by  Poincare.  The  order  of  a  term  being  a, 
the  rank  of  a  term  is  represented  by  a  -  m,  or  by  the  order  less  the  exponent 
of  t.  A  term  of  high  order  is  initially  small,  but  if  m  is  large  it  will  grow 
rapidly  in  importance,  so  that  ultimately  the  terms  of  the  lowest  rank  will 
have  the  greatest  significance. 

The  occurrence  of  long-period  terms  with  small  divisors  has  been  noticed. 
In  the  higher  orders  these  divisors  will  be  combined  and  raised  to  higher 
powers  by  the  subsequent  integrations.  Let  m'  be  the  sum  of  the  exponents 
of  such  divisors  in  any  term.  Then  the  class  of  that  term  is  defined  .by  the 
number  o  —  ^  (m  +  m').  It  will  now  be  clear  that  the  value  of  these  different 
categories  depends  on  the  length  of  time  contemplated.  For  relatively  short 


166, 167]  Absolute  Perturbations  183 

intervals  the  most  important  terms  are  those  of  low  order.  In  longer  intervals 
the  terms  of  low  class  rise  into  prominence.  And  finally  it  is  the  terms  of  low 
rank  which  have  the  greatest  influence  in  the  ultimate  destiny  of  the  system. 

But  here  'a  question  naturally  arises.  How  far  is  the  form  in  which  the 
terms  present  themselves  natural  to  the  problem,  and  how  far  are  they  the 
artificial  product  of  the  particular  method  by  which  they  are  obtained  ?  It  is 
evident  that  the  physical  importance  of  this  question  is  notr  quite  the  same 
in  all  cases.  Thus  a  mean  motion  in  the  position  of  the  node  or  perihelion 
may  be  admitted  without  any  serious  direct  consequences  to  the  nature  of  the 
system.  On  the  other  hand,  a  purely  secular  term  in  the  mean  distance  or 
the  eccentricity,  taken  by  itself  without  compensating  circumstances,  must 
ultimately  prove  fatal  to  the  stability.  The  general  problem  suggested  is 
very  difficult  and  the  reader  is  referred  to  the  first  volume  of  Poincar^'s 
Lecons  de  Mecanique  Celeste  for  a  thorough  discussion. 

It  must,  however,  be  pointed  out  that  the  form  of  the  results  may  be 
perfectly  legitimate,  so  far  as  it  goes,  and  at  the  same  time  not  in  any  way 
inconsistent  with  the  stability  of  the  system,  though  a  decision  is  beyond  the 
range  of  the  above  elementary  methods.  It  is  impossible  to  be  satisfied  with 
the  solution  here  described  as  a  final  representation,  and  this  feeling  is  ob- 
viously suggested  by  considering  the  mixed  terms.  Since  the  corresponding 
oscillations  increase  in  amplitude  indefinitely  with  the  time  the  departure 
from  the  original  configuration  will  become  so  great  that  the  fundamental 
assumption  of  small  displacements  in  forming  the  equations  for  the  variations 
will  be  contravened.  Then  one  of  two  things  will  happen.  Either  the  mutual 
forces  will  tend  to  restore  the  original  configuration,  and  there  will  be  stability, 
or  the  forces  will  tend  to  magnify  the  disturbance,  and  there  will  be  instability. 
But  in  either  case  equally  the  method  adopted  breaks  down  and  the  funda- 
mental question  remains  unanswered. 

How  then  are  the  statements  to  be  reconciled,  that  the  method — which 
is  the  method  on  which  the  existing  theories  of  the  major  planets  are  actually 
based — may  be  perfectly  legitimate,  and  that,  while  the  form  of  the  terms  to 
which  it  leads  obviously  suggests  instability,  complete  stability  is  never- 
theless entirely  possible  ?  The  simple  answer  is  that  it  is  only  necessary  to 
imagine  that  v  in  the  argument  of  any  term  is  itself  a  function  of  the 
disturbing  masses.  Now  the  above  method  involves  a  development  in  powers 
of  the  masses,  and  when  the  parameters  which  represent  the  masses  are  thus 
forced  out  of  the  circular  functions  they  carry  the  time  t  explicitly  with  them, 
and  the  appearance  of  secular  and  mixed  terms  is  a  natural  consequence. 
Yet  the  development  in  terms  of  the  masses  may  be  convergent  and  entirely 
legitimate.  In  this  way  it  will  be  seen  that  the  occurrence  of  secular  and 
mixed  terms  is  compatible  with  stability,  though  a  profound  discussion  is 
necessary  for  a  positive  conclusion  on  this  point. 


184  Absolute  Perturbations  [CH.  xv 

The  case  of  a  planet  moving  in  a  resisting  medium  is  quite  different. 
There  is  then  a  definite  loss  of  energy  and  the  effect  of  the  secular  changes  is 
not  doubtful. 

168.  In  the  theories  of  the  planets  on  which  the  existing  tables  have 
been  based  the  coordinates  of  the  planets  relative  to  the  Sun  have  been  used 
and  this  fact  governs  the  form  of  the  disturbing  function,  which  is  distinct 
for  each  pair  of  planets.  For  practical  purposes  this  choice  of  coordinates  is 
an  obvious  one.  But  for  theoretical  purposes  it  is  unsuitable,  chiefly  because, 
like  the  common  system  of  elliptic  elements,  it  is  ill  adapted  to  the  transfor- 
mations which  are  an  essential  feature  of  the  dynamical  methods  initiated  by 
Hamilton.  Another  system  of  coordinates,  due  to  Jacobi,  will  therefore'  now 
be  introduced. 

Let  (gi,  rn,  &)  be  the  coordinates  of  the  mass  TO;  in  a  system  of  n  masses 
m^mz,  ...,mn,  the  origin  being  any  fixed  point.  The  masses  are  taken  in 
any  fixed  order,  represented  by  the  suffixes,  which  is  quite  independent  of 
any  arrangement  which  may  be  visible  in  the  system.  Let 


Let  (X{,  Yit  Zi)  be  the  coordinates  of  the  point  Gi,  which  is  the  centre  of  mass 
of  the  partial  system  m1}  m2,  ...,  mi,  so  that 


/Z;_!  Zi_!,       &  =  X1  . 

Let  (xi,  yi,  z^  be  the  coordinates  of  mi  relative  to  Gi-l}  so  that 


Thus  (#2,  2/2.  z-i)  are  the  coordinates  of  ra2  relative  to  m1}  or  (£2—  £,,  rj^—^,  £2—£i)', 
(x3,  2/3,  23)  are  the  coordinates  of  m3  relative  to  G2,  the  centre  of  mass  of  m1}  m2  ; 
and  so  on.  There  are  no  coordinates  (xlt  ylt  z^).  By  the  above 


O;  -  Mf- 
Hence  on  eliminating  the  product  term 


and  on  addition  of  all  the  equations  of  this  type 


xflm  +  fj,nXn2. 

=  j= 

The  relations  between  the  coordinates   have   been-  written   down  for   one 
only.     But  they  are  linear  and  the  same  for  all  three  coordinates  separately. 


167-169]  Absolute  Perturbations  185 

Therefore  they  also  apply  to  the  velocities.    Hence  if  T  is  the  kinetic  energy 
of  the  system, 


i=2 


But  (Xn,  Yn,  Zn)  are  the  coordinates  of  the  centre  of  mass  of  the  system. 
They  are  absent  from  the  potential  function  and  are  in  fact  ignorable  coordi- 
nates. The  known  integrals  for  the  centre  of  mass  follow  immediately  and 
these  coordinates  can  be  suppressed.  The  problem  of  n  bodies  is  thus  reduced 
to  a  problem  of  n  —  1  fictitious  bodies  and  the  total  order  of  the  differential 
equations  of  motion  is  reduced  by  6. 

169.     The  new  form  of  the  areal  integrals  is  easily  found.     For 


-  (mZi  -  /H-i^i-i)  (pi*  4 
(pt  -  M,--,)2  (M  -  gifr)  =  tf  (  Yt  -  Yt-i 
and  hence 


The  sum  of  all  equations  of  this  type  gives 

»  .  .  . 

2  mi  ((ifcfc  -  ft  17*)  -  Pi-ilH~\(giii  -  *&)}  =(*n(YnZn-  Zn  Yn). 

i=l 

But  'it  is  possible  to  write  Xn  =  Yn  =  Zn  =  0  ;  that  is  equivalent  to  taking  the 
centre  of  mass  of  the  system  as  the  origin  of  the  coordinates  (£,  7jiy  ft).  Thus 
the  areal  integrals  now  take  the  form 


2  niim^jii  1  zi&i  —  XiZ?)  =  c2 


•O  j    /  •  •     \    

where  (c1}  c2,  C3)  are  the  angular  momenta  of  the  system  about  fixed  axes 
through  the  centre  of  mass.  The  direction  of  the  axes  has  remained  the 
same  throughout. 

Let  (ca,  c2,  cs)  be  considered  as  the  components  of  a  constant  vector  C, 
mi  fji-t  fj,i~l  (&i,  yiy  Zi)  as  the  components  of  a  vector  Mit  and  (#,-,  yit  z{)  as  the 


186  Absolute  Perturbations  [CH.  xv 

components  of  a  vector  r;.     Then  in  quaternion  notation  the  above  three 
integrals  may  be  represented  by  the  single  equation 


Hence  in  the  problem  of  three  bodies 


These  three  vectors  are  therefore  coplanar.  But  V(r2M%)  is  normal  to  the 
plane  of  r2,  M2,  that  is,  to  the  instantaneous  orbit  of  the  fictitious  planet  2. 
Similarly  V(rsM3)  is  normal  to  the  instantaneous  orbit  of  the  fictitious  planet  3, 
and  clearly  C  is  normal  to  the  invariable  plane.  Hence  the  nodes  of  the  instan- 
taneous orbits  of  the  two  fictitious  planets  on  the  invariable  plane  coincide. 

This  important  property  explains  the  so-called  elimination  of  the  nodes, 
which  in  an  explicit  form  is  due  to  Jacobi.  In  the  more  common  system  of 
astronomical  coordinates  it  disappears  from  view.  The  reader  who  is  un- 
acquainted with  the  elements  of  quaternions  will  have  no  difficulty  in  finding 
an  alternative  form  of  proof,  as  in  §  22. 

170.  The  body  denoted  by  1  will  now  be  identified  with  the  Sun,  and 
i  or  j  will  have  the  values  2,  .  .  .  ,  n.  The  potential  energy  of  the  system,  when 
the  units  are  chosen  so  that  the  constant  of  gravitation  is  unity,  is 

TT  —  —  S  mimi  _  V  mi>  mj 

'   AM  /     '    Aij 
where 


Also  the  kinetic  energy,  when  the  coordinates  (Xn,  Yn,  Zn)  are  ignored,  is  T, 
where 


Let 

1 

pi'1^,...,     H=T+  U. 


Then  the  equations  of  motion  of  the  system  may  be  written  (§  124) 
dxi  _  dH     dx^  _     dH 

~dt~d^'  ~dt'~  ~a^'   to**)- 

Now 

(l*i  -  /Xi_i)  &  =  piXi  -  /ii_!  Zf_!  = 

and  therefore 

Sf+i  ~  £i=  xi+\ 

Hence  by  the  addition  of  such  equations 


i69-i7i]  Absolute  Perturbations  187 

which  expresses  the  relative  coordinates  £  j  —  £,  ,  .  .  .  in  terms  of  the  coordinates 
%i,...,  and  shows  that  the  latter  differ  from  the  former  only  by  quantities  of 
the  first  order  in  the  small  masses.  In  particular,  for  the  body  2,  which  may 
be  identified  with  any  one  of  the  planets,  there  is  no  difference. 

Let  V  be  reduced  to  its  terms  £/,  of  the  lowest  order  in  the  small  masses, 
which  is  the  first.     Then 


for  Ti  differs  from  A1(i  by  a  quantity  which  involves  the  masses.    The  equations 
of  motion  reduce  to 

_  dHi       „  =  T     jj 

=  ' 


dt  ~  dxi'  '     dt 
or  in  more  explicit  form 

fJLi_1Hi-lxi  =  -  m.,  Xi/ri3,     (x,y,  z). 
These  are  the  equations  of  undisturbed  elliptic  motion,  and  in  particular 

x2  =  -  (m,  +  m2)  x2fr23,     (x,  y,  z) 

which  agree  naturally  with  the  usual  equations  of  a  planet  relative  to  the 
Sun  in  undisturbed  motion,  and  give  a  mean  distance  a2  with  the  usual 
meaning.  For  the  other  bodies  the  equations  are  of  the  same  form  and  have 
precisely  similar  solutions,  but  the  elements  o^  will'  differ  from  the  ordinary 
elements  slightly  because  (set,  yi,  Zi)  are  not  coordinates  relative  to  the  Sun 
unless  i  =  2.  This  is  not  material  to  the  purpose  in  view  because  the  body  2 
represents  any  planet  and  any  proposition  which  is  proved  for  it  must  be  true 
generally. 

171.  These  equations  for  the  undisturbed  motion  can  now  be  solved  in 
terms  of  canonical  constants.  When  the  latter  are  treated  as  variables,  they 
satisfy  canonical  equations  formed  with  R=U1  —  U.  As  in  §  143  this  value 
of  R  may  be  modified  by  adding  2  mfj?/2L'2,  where  m  =  m;/^//^  and 
p  =  mifii/  m-i  in  view  of  the  explicit  form  of  the  undisturbed  equations.  Then 
any  of  the  different  sets  of  variables  explained  in  that  section  can  be  used, 
and  the  last  set,  now  denoted  by  (L1,  £/,  £/;  X,  ^  ,  ij2'),  will  be  chosen.  The 
equations  for  the  perturbations  can  now  be  written 


dt       d\i  '        pi       dt 


Pi        dt      df)i  '        /if        dt 
where 

V  =  -  U  +  U,  +  m* 


There  are  n  —  1  pairs  of  equations  in  (Li,  \i)  and  2(n  —  1)  pairs  in  (£/,  ?;/), 
but  there  is  no  need  here  to  distinguish  between  the  eccentric  and  oblique 


188  Absolute  Perturbations  [CH.  xv 


variables.     From  this  point  the  former  use  of  (ft-,  iji,  &)  as  the  rectangular 
coordinates  of  m^  disappears. 

A  little  explanation  may  be  necessary  to  account  for  the  appearance  of 
the  mass  factors  of  the  momenta  x{  in  the  equations.  In  §  135  giving  the 
Hamilton-Jacobi  solution  for  undisturbed  elliptic  motion  the  single  factor  m, 
representing  the  mass  of  the  moving  body,  was  removed  consistently  from  U, 
Tand  H.  Similarly  in  §  139  U—R  was  written  in  the  place  of  U,  R  being 
the  disturbing  function  in  its  common  form,  whereas  the  true  increment  in 
the  potential  energy  is  —  mR.  But  here  it  is  not  possible  to  divide  the  more 
general  function  U  —  U^  as  a  whole  by  any  particular  mass,  though  it  is 
possible  to  do  so  as  regards  the  set  of  equations  corresponding  to  a  particular 
value  of  i.  Hence  it  was  necessary  to  restore  the  mass  factors  in  the  manner 
shown.  But  now  they  can  be  removed  by  the  change  of  variables, 


N 


and  the  equations  then  become 


_ 
dt  ~d\i'       dt  ~      3L, 

d!ji=dV        dm=_dV 
dt      dr)i'        dt          dgi, 
where 

V=  -  U  +  U,  +  m,2  2  mffi 

The  terms  added  to  U-^  —  U  depend  on  the  Li  only,  and  affect  one  type  of 
equation,  namely 


so  that  \i  =  ntt  +  h  and  nt  is  the  mean  motion  in  the  preliminary  solution. 
The  first-order  perturbations  of  Xt-  will  require  the  first-order  perturbation  of 
Li  to  be  included  in  the  term  from  which  wt-  originates. 

172.  It  is  not  at  present  very  necessary  to  consider  in  detail  the  form  of 
expansion  of  U—  U^.  It  can  in  the  first  place  be  expanded  in  powers  and 
products  of  the  small  masses  m.{  and  of  the  coordinates  (#,-,  y,-,  2,-).  The  latter 
can  be  expanded  in  powers  of  Li,  ^,  77$  with  purely  periodic  functions  of  X,:. 
Hence  U—Ul  can  be  expanded  in  the  same  form,  and  arranged  in  orders  of 
the  masses,  beginning  with  the  second  since  the  first  has  been  removed  by  Ul  • 
Thus  if  the  fourth  order  in  V  be  neglected,  V=V2+V3,  where  F2  is  of  the 
second  order  and  Vs  of  the  third,  and  V2  contains  at  most  two,  Vs  at  most 
three,  mean  longitudes  X;  in  its  arguments,  the  coefficients  of  the  periodic 
terms  being  rational  and  integral  functions  of  Li,  £;,  77*. 


171,  172]  Absolute  Perturbations  189 

The  perturbations  of  the  first  order  can  now  be  obtained  in  the  usual  way 
by  neglecting  F3  and  substituting  initial  values  of  Li}  &,  ?/;  in  F2,  including 
i^t  4-  \i°  for  A;.  This  process  gives 

Li  =  L{°  +  ^Li°,  \  =  mt  +  \i°  +  8,  v,  &  =  &°  +  5!  &°,  77*  =  <  +  ^  V 
where  Zf°,  .  .  .  are  constants  and  S^Lf,  .  .  .  are  the  perturbations  of  the  first  order. 
Owing  to  the  form  of  F2,  9F2/9Af  is  purely  periodic  and  free  from  any  term 
independent  of  \{.  Hence  ^X/  is  also  periodic  and  free  from  a  secular  term. 
But  the  other  elements  will  contain  a  term  multiplied  by  t,  arising  from  the 
terms  independent  of  \{  in  the  partial  derivatives  of  F2,  together  with 
periodic  terms.  To  the  second  order  let 

ln-Lf+blf  +  SiLt. 

In  F3,  which  must  now  be  retained,  it  suffices  to  substitute  the  constant 
values  Li0,...  for  Li,...,  and  r^-fA/  for  A^;  but  in  F2  it  is  necessary  to 
substitute  Li°  +  S1Li°,  ...  for  Li,...,  though  only  the  first  powers  of  these 
perturbations  are  required.  Hence  the  equation 

^  (L?  +  B,  L?  +  S2  If)  =  ~  (  F2  +  F3) 
gives,  when  account  is  taken  of  the  solution  for  the  first  order, 


By  the  same  argument  as  applied  to  F2  in  the  first  approximation  the  last 
term  gives  rise  to  periodic  terms  only.  Hence  a  search  for  secular  terms  can 
be  confined  in  the  first  place  to  the  expression 


_I,dt+  i2^  [*r*dt-  82*JL  f 
dLf  a(  +  ax,9f/  J  a^  a{    diidrjf  J 


Here  the  multipliers  of  the  integrals  are  all  purely  periodic,  owing  to 
differentiation  with  respect  to  Af.  The  integrals  themselves  contain  secular 
terms  in  t.  Hence  on  integration  the  products  will  give  rise  to  periodic  and 
mixed  terms,  but  not  to  purely  secular  terms  on  this  account.  The  latter  must 
arise,  if  at  all,  from  a  constant  term  in  the  products.  The  only  way  in  which 
this  could  happen  would  be  connected  with  terms  in  the  development  of  F2  of 
the  form 

F2  =  B  sin  (ki\{  +  kj\j)  +  C  cos  (ki\i  +  kj\j)  =  B  sin  yfr  +  C  cos  i/r. 
But  for  these 


2  ra_F2  ,,_JH^  f 

f  J  ax,-         d\id\j  1 


+  k^ 
=  0. 


190  Absolute  Perturbations  [CH.  xv 

In  a  similar  way  those  terms  which  might  produce  constant  terms  neutralize 
one  another  between  the  other  pairs  of  products  and  therefore  no  purely 
secular  part  of  8.2Li°  can  arise  in  this  way. 

But  the  above  expression  is  not  complete,  because  SjX,/  depends  on  SjZ/ 
as  well  as  on  F2.     For,  by  the  last  equation  of  §  171, 
dSjX,-0         3F2     37-2^3... 


dt  dLf 


^  *  7  o 
l   j 


so  that  there  is  an  additional  part  of  82Li(l  n°t  yet  considered.     It  is  given  by 


where  J.  is  a  constant.     But  terms  in  F2  of  the  above  type,  taken  in  the  form 
D  sin  (i/r  +  h),  lead  to 

It  <«w  =  A  •  **>  D  sin  (*  +  A>  •  (kj£w  D  cos  <*  +  A> 

d  Z..J..2 

-2 


Therefore  this  part  of  82Li°  is  purely  periodic. 

Hence  there  are  no  purely  secular  terms  in  82Li°,  a  proposition  which 
Poincare'  has  proved  in  the  more  general  form  :  there  are  no  purely  secular 
perturbations  of  Li  in  any  order  of  rank  lower  than  2. 

This  applies  in  particular  to  L.2,  But  «2  =  ML./,  where  M  is  a  constant 
mass  factor.  Hence 

a2  +  &ia2  +  S.2a2  =  M  (Lz  +  ^L^  4-  82L2)2 
B.a,  =  2ML281L2,     $,a.2  =  M  {(S.L^  +  2L2  (S2Za)} 

the  affix  °  being  now  omitted.  But  §1L2  is  purely  periodic,  and  82Z/2  has  no 
purely  secular  term.  Hence  to  the  second  order  in  the  masses  there  is  no 
secular  inequality  in  the  mean  distance,  for  it  has  been  remarked  that  a2 
represents  the  mean  distance  of  any  of  the  planets.  This  is  Poisson's  theorem, 
an  extension  of  Laplace's  corresponding  theorem  for  the  first  order,  and  it  is 
the  most  important  elementary  result  bearing  on  the  stability  of  the  solar 
system. 

173.  On  the  other  hand  there  are  evidently  mixed  terms  of  order  2  and 
rank  1  in  Li.  Hence  the  existence  of  purely  secular  terms  of  order  3  and 
rank  2  in  a2  can  be  anticipated.  For  even  without  pushing  the  approximation 
further  and  examining  8SL2  it  is  obvious  that  ^M^L^.  82L2  constitutes  a  part 
of  8sa.2.  Therefore  the  combination  of  a  term  A  cosmt  in  81L2  with  a  term 
Btcos  mt  in  82L2  will  give  a  term  MABt  in  83a2.  Such  terms  were  first  shown 
to  exist  by  Spiru-Haretu  in  1876. 


172,  i7s]  Absolute  Perturbations  191 

On  one  condition  true  secular  inequalities  of  the  first  order  occur  in  the 
mean  distances.     Since 


U  —  £/!  =  S  A  cos  (ki\i  +  kj\j  +  h) 
to  its  lowest  order, 

a  V/d\i  =  2  Aki  sin  (&A;  +  kfc  +  h). 

For  perturbations  of  the  first  order  the  coefficients  are  constants  and  \  -  n^t, 
\j  —  Hjt  are  also  constant.  Hence 

dLi/dt  =  ^Aki  sin  (mt  -f  h'). 

A  constant  term  results,  producing  a  secular  inequality,  if  m  =  &iW;  +  kjitij  =  0, 
which  is  possible  only  if  nif  rij  are  commensurable.  This  possibility  was 
considered  in  the  previous  form  of  discussion  and  excluded.  But  it  is  in  effect 
ruled  out  by  its  own  consequences.  For  if  a  body  were  artificially  or  fortui- 
tously projected  in  such  a  way  as  to  have  a  mean  motion  commensurable 
(e.g.  £,  i,...)  with  the  mean  motion  of  a  disturbing  body,  its  mean  distance 
would  be  subject  to  a  secular  disturbance  from  the  beginning,  and  therefore 
the  commensurability  of  its  motion  would  be  definitely  destroyed.  Hence  if 
the  minor  planets  be  arranged  in  order  of  distance  from  the  Sun,  it  is  to  be 
expected  that  gaps  will  be  found  in  the  frequency  at  distances  corresponding 
to  mean  motions  commensurable  with  that  of  Jupiter,  and  it  is  so.  And 
similarly  divisions  in  the  rings  of  Saturn  can  be  attributed  to  the  secular 
perturbations  of  the  constituent  meteoric  bodies,  produced  by  the  commen- 
surable motions  of  any  satellite  which  may  be  effective.  This  also  has  been 
verified  for  the  action  of  Mimas  by  Lowell  and  Slipher. 

Nevertheless  among  the  many  minor  planets  a  few  are  naturally  found 
whose  motions  are  nearly  commensurable  with  Jupiter's  mean  motion.  For 
these  the  long-period  terms  with  small  divisors  are  highly  important,  and  the 
terms  of  low  class  play  a  far  larger  part  than  in  the  theories  of  the  major 
planets.  The  special  difficulties  thus  presented  require  special  methods  of 
treatment,  and  such  have  been  suggested  by  JJansen,  Gylden  and  others. 
Poincare  has  used  an  application  of  the  principle  of  Delaunay's  method.  The 
proper  treatment  of  this  class  of  minor  planets  presents  perhaps  the  most 
interesting  problems  to  be  found  in  dynamical  Astronomy  at  the  present 
time. 


CHAPTER  XVI 


SECULAR    PERTURBATIONS 

174.  In  the  preceding  chapter  it  has  been  shown  that  the  mean  distances 
in  the  planetary  system  are  free  from  purely  secular  inequalities  when 
developed  to  the  second  order  in  the  masses.  The  general  nature  of  the 
secular  perturbations  in  the  other  elements  will  now  be  examined.  It  may 
be  convenient  to  modify  slightly  the  equations  obtained  in  §§  170,  171.  By 
reducing  U  to  its  terms  of  the  lowest  order  the  equations  of  motion  there 
took  the  explicit  form 

p-i^^Xi  =  -  m^rf,  (x,  y,  z) 

which  are  satisfied  by  the  osculating  motion  of  a  planet,  according  to  its 
ordinary  definition,  when  i  =  2,  but  not  otherwise.  But  if  £//  be  substituted 
for  Ult  where 

•  Ui  =  -  2  (wj  +  nn)  mmi-i/ Wi 

a  form  which  will  be  found  to  differ  from  U^  by  terms  of  the  third  order 
only,  the  explicit  equations  of  motion  become 

Xi  =  -  (m,  +  mi)  Xi/n3,  (as,  y,  2) 

which  are  the  ordinary  equations  in  the  undisturbed  problem  of  two  bodies, 
and  are  satisfied  by  the  osculating  elements  taken  in  their  usual  sense.  The 
mass  factors  of  the  momenta  are  as  before  m^i^//^,  but  the  constants  of 
attraction  are  p,  =  ml  +  m^  Hence  the  equations  for  the  variations  will  now 
be  based  on 

V  =  -  U+  US  +  2  (nh 

=  -U+U1'  +  ^  (m, 

The  relation  between  Zt  and  L{  is  the  same  as  before,  but  the  meaning  of 
both  is  changed  (except  when  i  =  2)  in  such  a  way  that  Li  bears  generally 
the  same  form  of  relation  to  di,  the  osculating  mean  distance  in  its  ordinary 
sense,  as  L2  to  o^. 


174,  175]  Secular  Perturbations  193 

Thus  the  transformations  of  §  143  give,  with  those  of  §  171, 

Li  =  (m-i  +  miY  at",      Gt  =  L{  cos  fa,  Hi  =  G{  cos  i 

k  =  €i  —  nTi  +  nit,         gi  =  iffi  —  Q,i,  hi  =  £li 

pit  1  =  2Z//  sin2  %fa,     piy  2  =  2Z/  cos  fa  sin2  £tj 

\  =  €i  +  nit,  &>i,i  =  —  ^i,  «<>t,  2=  —  ^f 

£;  =  w;  (w!  +  w;)  V;_!  pi"1  a? 

£,:>  j  =  2Zf*  sin  ^fa  cos  ta^,  77,;,  j  =  -  2£,;2~  sin  £<£;  sin  tsr£ 

^  2  =  2Z^  cos*  <fo  sin  %it  cos  ft;,     ^  2  =  —  2J^*  cos*  <£t-  sin  ^  sin  nf. 

Here  sin  fa  =  &i  and  no  confusion  is  possible  between  the  inclination  i  and 
the  subscript  i,  which  is  merely  a  distinguishing1  mark  for  the  several  planets. 

175.  It  is  obvious  that  U  —  U-{  can  be  expanded  in  powers  of  Xi  —  aiy 
yi  -  bi,  Zi  -  Ci  where  (a^,  bi}  d)  are  what  (xi}  yi}  ^)  become  when  ff  =  r)t  =  0. 
Now  (§65)  the  heliocentric  coordinates  are  generally 

x  =  r  cos  O  cos  (w  +  *CT  —  H)  —  r  cos  t  sin  fl  sin  (w  +  CT  —  O) 
=  r  cos2  ^t  cos  (w  +  OT)  +  r  sin2  \i  cos  (w  +  -or  —  2H) 

y  =  r  sin  fl  cos  (w  +  VT  —  fl)  +  r  cos  *  cos  H  sin  (w  +  w  —  O) 
=  r  cos2  \i  sin  (w  +  zr)  —  r  sin2  \i  sin  (w  +  CT  —  2O) 

^  =  r  sin  i  sin  (w  +  ^  —  O) 

w  being  the  true  anomaly.     Let 

X  =  r  cos  (w  -  M),     Y  =  rsin(w-M),     if  =  A,  --or 
M  being  the  mean  anomaly.     Then 

x  =  X  {cos2  1  1  cos  X  +  sin2  £i  cos  (X  —  2ft)} 
-  F  {cos2  %i  sin  X  +  sin2  $i  sin  (X  -  2ft)} 

y  =  X  {cos2  £i'  sin  X  —  sin2  \i  sin  (X  —  2ft)} 
+  F{cos2  \i  cos  X  -  sin2  \i  cos  (X  -  2ft)} 

z  =  X  sin  t  sin  (X  —  ft)  +  Fsin  i  cos  (X  —  ft). 

The  coefficients  of  X  and  F  here  involve,  besides  periodic  functions  of  X,  the 
quantities 

cos2  \it     sin2  \i  cos  2ft,     sin2  \i  sin  2ft,     sin  i  cos  ft,     sin  i  sin  ft 

and  since 

g*  +  rj,2  =  4>L  sin2  £  fa     &  +  7722  =  4£  cos  <£  sin2  $  i 

tan  «r  *«  —  i,  tanft  =  — 


it  is  easily  verified  that  the  five  quantities  can  all  be  expanded  in  powers  of 
£1,  i?i,  &,  »?2.     Also 

r  cos  w  =  a  (cos  E  —  e),     r  sin  w  =  a  cos  </>  sin  .fi* 

P.  D.  A.  13 


194  Secular  Perturbations  [CH.  xvi 

E  being  the  eccentric  anomaly,  and  therefore 
X/a  =  —  e  cos  M  +  cos2  -£<£  cos  (E  —  M) 

+  \  sec2  %(f)  {e"  cos  2M cos  (E-M)-  e*  sin  2M  sin  (# -  Jlf )} 

Yfa  =es'mM+  cos2  ^  sin  (E  -  M) 

-  i  sec2  ^  {e2  cos  2Jlf  sin  (E  -  M)  +  e2  sin  2ilf  cos  (#  -  M)} 

which  are  forms  easily  verified.  Since  cos3^<£,  sec2^</>  can  be 'expanded  in 
terms  of  e2  =  sin2  </>,  these  forms  show  that  X,  Y  can  be  expanded  in  powers 
of  e  sin  M,  e  cos  M  if  this  is  true  of  sin  (E  —  M),  cos  (E  —  M).  But  Kepler's 
equation  may  be  written 

6  —  x  cos  6  —  y  cos  6  =  0,     6  =  E  —  M,     x  =  esmM,     y  =  e  cos  M 

and  0  can  be  expanded  in  powers  of  x,  y.  Hence  sin  (E  —  M),  cos  (E  —  M) 
can  be  expanded  in  powers  of  e  sin  M,  e  cos  M,  and  therefore  also  X  and  F. 
But  this  shows  that  X,  T  can  be  expanded  in  powers  of  e  sin  •nr,  e  cos  •sr  with 
coefficients  involving  periodic  functions  of  X,  since  M  =  X  —  -ST.  And  e  sin  OT, 
e  cos  w  can  be  expanded  in  powers  of  £1?  T/I}  as  can  easily  be  seen,  with 
coefficients  involving  L.  Hence  (x,  y,  z)  can  be  developed  in  powers  of 
tri>  %>  £2,  *?2  with  coefficients  involving  L  and  periodic  functions  of  X.  There- 
fore finally  U—  f/i'  can  be  expanded  in  powers  of  £i(1,  17^,  £fi2,  77^  2  with 
coefficients  involving  Z;  and  periodic  functions  of  \if  and  the  supplementary 
part  of  V  involves  LI  only. 

It  is  assumed  that  the  inclinations  of  the  orbits  are  very  small.  Now 
there  are  two  ways  of  regarding  retrograde  motion  in  an  orbit  whose  plane 
differs  little  from  the  orbits  of  planets  moving  in  the  opposite  sense.  It  is 
possible  to  take  the  mean  motion  nt  as  positive.  Then  the  inclination  is 
near  TT  and  is  not  small.  Or  it  is  possible  to  take  the  inclination  as  small, 
and  to  regard  n{  as  negative.  Then  since  n^Lf  is  a  positive  mass  function, 
Li  is  negative  and  therefore  £f,  T/J  are  imaginary.  All  the  orbits  will  therefore 
be  supposed  to  be  described  in  the  same  (direct)  sense,  which  is  true  of  the 
planetary  system  but  not  always  of  the  satellite  systems. 

This  remark  has  an  obvious  bearing  on  theories  of  cosmogony.  For  if 
high  inclinations  and  in  particular  retrograde  motions  were  unstable,  such 
forms  of  motion  would  not  be  permanently  maintained.  Now  the  nebular 
hypothesis  of  Laplace  is  very  largely  based  on  the  observed  fact  that  the 
planetary  motions  are  nearly  coplanar.  If,  however,  such  a  type  of  motion 
is  alone  stable,  the  observed  fact  loses  its  significance  in  this  connexion  and 
no  deduction  of  the  kind  is  to  be  drawn  from  it.  The  question  of  stability 
in  general,  beyond  the  range  of  inclinations  to  be  found  in  the  actual  planetary 
system,  is  therefore  important,  though  beyond  the  range  of  this  work. 


175-177]  Secular  Perturbations  195 

176.     When  the  secular  part 


[-  U+ 

'  'i,  2 


which  is  free  from  \  is  considered,  certain  properties  of  the  development  are 
easily  seen.  For  this  being  independent  of  the  direction  of  the  chosen  axes, 
the  substitutions 


f  ,  2  >          'i,  a 


are  all  possible  without  affecting  the  result.  Thus  (a)  follows  when  £lit  -&i 
are  altered  by  TT,  or  when  the  axes  of  xy  are  rotated  through  TT  in  their  own 
plane.  Similarly  (6)  follows  when  this  rotation  is  made  through  ^TT.  Again 
(c)  is  produced  when  flf  (but  not  -or;)  is  altered  by  TT,  and  this  is  equivalent 
to  reversing  the  axis  of  z.  Finally  (d)  is  obtained  by  changing  the  signs  of 
all  the  angles  X;,  fit-,  wt-,  which  is  equivalent  to  reversing  the  axis  of  y.  The 
change  in  X,:  is  of  no  further  importance  here  since  X;  is  absent  from  the 

terms  now  considered. 

« 

Certain  properties  of  the  exponents  in  the  expansion  are  now  obvious. 
For  2  (pl  +  ql  +p2  +  q2)  must  be  an  even  number  to  satisfy  (a),  and  S  (pz  +  Qz) 
to  satisfy  (c).  Hence  '2t(p1  +  ql)  is  also  an  even  number.  Similarly  (d) 
requires  S  ((ft  +  #2)  to  be  even,  and  therefore  2  (  p^  +pz)  must  be  even.  Hence 
in  the  second  degree  there  can  be  no  terms  of  the  form  £77  or.  ££2,  17^2- 
But  if  terms  of  the  fourth  degree  be  neglected,  only  terms  of  the  second 
degree  involving  £,  77  remain.  These  terms  can  therefore  be  written  down  in 
the  form 

[-U+  Uf]  =  %%A  it  j  (&,  !  &  j  +  i?*,  i  "nj,  i)  +  2  i-Bi,  j  (&,  2  1>-,  2  +  i\i,  2  *)j,  2) 
where  the  coefficients  of  &&,  rjiVj  are  taken  to  be  the  same,  both  for  the 
eccentric  and  the  oblique  variables,  in  accordance  with  the  substitution  (6), 
and  terms  £i|y,  17^-  are  reckoned  twice  when  i,j  are  different,  but  Ai>j  =  Ajii> 
Bi,j=  B/,t- 

177.  It  will  be  of  interest  to  obtain  the  explicit  values  of  A{j,  St  j  for 
the  lowest  order  in  the  masses.  The  principal  part  of  the  disturbing  function 
is  SmiWjA"1  and  it  has  been  seen  in  §  159  that  the  complementary  part 
contains  periodic  terms  only.  The  distances  At-(  j  involve  coordinates  (#;,  y^  zj) 
which  themselves  contain  the  masses.  But  to  the  lowest  order  these  coordi- 
nates are  identical  with  the  relative  coordinates  commonly  in  use,  and  the 
methods  of  Chap.  XIV  can  therefore  be  employed.  Two  planets,  1,  2,  will  be 
first  considered.  Then  in  the  notation  of  §  152,  when  the  orbits  are  circular, 
a2  A-1  =  6°'  °  =  |  b£  -  %avb£  +  .  .  . 

13—2 


Secular  Perturbations  [CH.  xvi 

with  the  exclusion  of  all  periodic  terms.  The  triangle  formed  by  the  two 
orbits  and  the  ecliptic  gives 

cos  J=  cos  i-i  cos  i2  +  sin  t\  sin  z'2  cos  (1^  —  fla) 
or  to  the  second  order  in  i1}  i2, 

v  =  sin2  ^J=  i  [i\  +  i-?  —  ^i\iz  cos  (Oj  —  fl2)}. 

Since  v  is  of  the  second  order  the  Laplace's  coefficient  bs1  is  derived  imme- 
diately from  the  circular  motion.  But  &i°  must  be  modified  to  include  the 
eccentricities,  the  orbits  being  now  treated  as  coplanar.  Let 

A02  =  «j2  +  a./  -  2^02  cos  6,     6  =  t^  -  nr2  +  Ml  -  Mz. 
Then  in  the  notation  of  §  157, 


A-i  =  -      exp.  \(Wl  -  M,}  A  +  (w2  -  M2)  A}  AT' 

\O%f       V^a/ 
and,  by  (22)  of  §  40  and  (30)  of  §  41, 

r/a  =  1  +  |e2  -  e  cos  M  -  ^e2  cos  2M  +  ... 

w  -  M  =  2e  sin  M  +  f  e2  sin  2M  +  .... 
Hence 

(a-ir)^  =  1  -  e  cos  M  .  D  +  \&  (1  -  cos  2M)  D  +  £e2j(l  +  cos  2Jf)  .  D  (D  -  1) 
exp.  (w  -  M}  D  =  1  +  2e  sin  M  .  D  +  f  e2  sin  2M  .  D  +  e2  (1  -  cos  2M)  .  D2. 

All  operating  terms  which  do  not  combine  M  l  ,  M2  in  the  form  M:  —  M2  will 
clearly  produce  periodic  terms  only.  And  terms  already  of  the  second  degree 
are  combined  with  no  others.  Therefore,  when  ineffective  terms  are  omitted, 
since  Dl  +  !Z)2  =  —  1, 

A-1  =  (!-<?!  cos  Ml  .  A  -  \e?  .  D,D2)  (1  -  ez  cos  M2.D2-  \e?  .  A  A) 

(1  +  2e,  sin  MI  .  D3  +  e?  .  A2)  (1  +  ^  sin  M2  .  D,  +  e?  .  A2)  Ar1 
=  {1  +  £<?i<?2  cos  (Ml  -  M2)  .  A  A  +  2^02  cos  (M1  -  M2)  .  A  A 
—  ei62  sin  (Mz  —  MJ  .  A  A  -  ei^  sin  (^i  ~  -^2)  •  A  A 

-  i  (^2  +  e22)  A  A  +  ^2  .  A2  +  e22  .  A21  Ao-1 

where    again    terms    involving    M1}   M2    or   J/j  +  M2    are    omitted.      Now 

A  =  —  A  =  9/5^  and,  since  a  =  a^a^ 

A  A^o""1  —  aia2  cos  6  .  A0~3  +  3  (a^  —  ajtta  cos  ^)  (a22  —  ^1^2  cos  ^)  A0~5 

=  a2~1  {a  cos  0  .  a23A0-3  +  3a  [f  a  -  (1  +  a2)  cos  0  +  £a  cos  20]  a25  Ar5} 
A2  Ar1  =  A2  Ao-1  =  -  A  A  A0~l  =  -«!«,  cos  ^  .  A0~3  +  3ai>a<?  sin2  0  .  A0-5 

=  ar1  {-  acos0.  a23  A0~3  +  fa2  (1  -  cos  20)  .  a25  A0-5} 
A  A  Ao"1  =  «ia2  sin  0  .  A0~3  —  Sa^  sin  0  (a^  -  a-^a^  cos  0)  A0~3 

=  ar1  {a  sin  0  .  a23  A0-3  -  3a2  (a  sin  0  -  \  sin  20)  a25  A0~5} 
D2D3\~l  =  —  a^a2  sin  0  .  A0~3  +  3aia2  sin  0  (a22  —  c^a-j  cos  0)  A0~5 

=  a-2-1  {-  a  sin  0  .  a23  Afl-3  +  3a  (sin  0  -  ^a  sin  20)  a25  A0-s|. 


177,  i7s]  Secular  Perturbations  197 

For  the  secular  terms  it  is  possible  to  write 

COS  (M!  —  M2)  =  COS  (0  —  W1  +  «r2)  =  COS  0  COS  (wj  —  -BT2) 

sin  (M1  -  M2)  =  sin  (0-^  +  tj8)  =  sin  0  cos  fa  -  <srz) 
since  sine  terms  and  cosine  terms  must  combine  separately. 

178.     The  secular  terms  of  the  second  degree  in  the  eccentricities  can 
now  be  written  down  in  terms  of  Laplace's  coefficients  (§  147)  thus  : 

/•A        rr*  -4—  -4-  £   f*    C*OS  \  "CT     .-_--  TTT    |      rj  — ^ 


+  «  (  V  -  V)  -  3a2  C«  (  V  -  V)  -  i  (  V  - 
+  «  (V  -  V>  -  3a  KV  -  ¥)  -  ^a  (^t1  - 

-  i  C^2  +  e?}  .  ar1  [a^1  +  3a  [|«6^  -  (1  +  a2)  &51 
+  i  (^  +  e£)  .  arl  {-  c^1  +  fa2  (6f°  -  6f2)j. 

To  simplify  this  expression  the  recurrence  formulae  (4)  and  (5)  of  §  148  with 
j  =  0  are  available  : 

(i-s+  1)  abl+l  -i(l+  a2)  6^  +  (i  +  s  -  1)  ab^1  =  0 

(i+S)bi=S(i+*)bi+1 

Thus 


+  f  «V)  -  taV  =  f«( 

and  the  last  line  of  the  expression  disappears.     Again 


=  f  {(1+  a2)  &f> 

—  l<i+*) 

Hence  the  penultimate  line  of  the  expression  reduces  to 


which  represents  all  the  terms  in  e}2,  e22. 

The  coefficient  of  +  |exe2  cos  (nr1  —  •sr2)  az~la.  is 

+  IV  +  i  V  -  f  (1  +  O  V  +  ¥a  V  +  I  (1  +  «2)  V  +  l« 
=  i  V  -  ¥a  V  +  f  (1  +  «2)  V  +  I  V 
=  i  V  -  if  [2  (1  +  a2)  6,2  -  tabf]  +  |  [(1  +  a2)  V  +  *a  6 


198  Secular  Perturbations  [CH.  xvi 

and  the  whole  of  this  term  is  therefore 

—  \e\e^  cos  (CTJ  —  CT2)  .  crafts2. 

Hence  the  terms  of  the  second  degree  in  the  eccentricities  and  inclinations 
for  two  planets  give  finally 

[A-1]  =  arX  \\  (e?  +  e?)  bS  -  \e,e,  cos  fa  -  ^2)  bf] 

-  lo^rX  {%2  +  Vs  -  2M2  cos  (H!  -  n2)}  6s1. 
But  to  this  order  (that  is,  neglecting  the  third  order  in  e,  i) 
%!  =  eli  cos  -nr,     f]1  —  —  eL*  sin  «r 
£2  =  iL*  cos  fl,     t]2  =  —  iL?  sin  O. 

By  translating  from  one  system  of  variables  to  the  other  and  taking  the  sum 
for  each  pair  of  planets,  it  follows  that 

[_  U+  Z7i']  =  iSmiro,  |(^  +  ^  +  ^  +  ^)  ^  (a,,  a,-) 

'    (\  JL/f  Lii  -Lj  LIJ  J 

2 


(&,  i  &,  i  +  'm,  ity,  i) 


[fc2  2  £2  2  O  /£          £  i  \~1  ^ 

C  {,    v         ^?t2          3?2         ^7?2  \  a  1    2  s  7    2     *"  ^1    2  V?    2/   I     Ti     /  \  / 

B1(a<,%)  =  ^6f»^.-.£ 

tc/y  \  tt-^  /  /i   J  0 


where 


o  (a.*  +  a/  -  2^^-  cos  ^^ 

»  /     \   a^  2  ^aA   2  r     aiaj  cos  2^  •  de 

02(0*1,  a,j)  =  -^  os  I  —  I  =  -  -  i  . 

«/    r  Vttj/      TT  J  o  (a.2  +  aj2  _  2a.a.  cos  5/)^ 

The  coefficients  of  Laplace  are  positive.  Therefore  the  quadratic  terms  in 
the  oblique  variables  are  a  negative  definite  form.  Further,  by  the  recurrence 
formulae, 

0  =  fats1  -  2  (1  +  a2)  bf  +  f  abf 

4 

Therefore 
But 

and  therefore 
which  shows  that 


Hence  the  quadratic  terms  in  the  eccentric  variables  are  a  positive  definite 
form. 


178,  179] 


Secular  Perturbations 


199 


179.  The  problem  of  the  small  eccentricities  and  inclinations  of  the 
planetary  system  is  now  brought  within  the  range  of  the  general  theory  of 
small  oscillations  about  a  steady  state  of  motion.  Indeed  a  knowledge  of  the 
principles  of  this  theory  shows  at  once  that  the  variations  in  the  eccentricities 
and  inclinations  are  periodic  and  stable,  for  this  follows  from  the  definite 
(positive  or  negative)  forms  of  the  quadratic  terms. 

Since  (§  176) 

[-  U+  Ul']  =  ^^Aij(^i^jjl  +  r)i!lr}jtl) 
the  corresponding  canonical  equations  are 


v  4  . 
* 


dt 

*K  2 

--     - 


forming  two  distinct  sets  of  linear  equations  with  constant  coefficients.  The 
results  will  clearly  be  of  the  same  general  kind  for  both,  and  it  is  only 
necessary  to  consider  the  eccentric  variables. 


Let  the  linear  transformations 


be  orthogonal,  so  that 


Vi  =  2  a,-, 


1  =  2a2t,  j,       0  =  2ofjOi,  t,     (  j  =}=  A;). 


Thus 


2  &  din  =  2  2  2 

i    j    k 


=  2  pt  dqt 
which  shows  that  such  a  transformation  is  also  canonical.     Now  let 

Then 


is  an  expression  which  is  independent  of  p^.     Therefore,  product  terms  being 
reckoned  twice, 


This  is  an  identity,  satisfied  by  all  values  of  £f.     Hence 

itjajik  -akaijk=  0 


200  Secular  Perturbations  [OH.  xvi 

and  this  system  of  equations,  for  the  values  i  =  2,  3,  ...,  n,  gives  a  consistent 
solution  for  a^  *,  provided  or*  is  a  root  of  the  equation 

•"•2,2  —  &  -"-2,3  -"2,4  •••        ==  "• 

-"•3,2  -"-8,  3  ~   tt  -"-8,  4 

-"•4,2  -0-4,3  -«4,  4~° 

This  is  a  symmetrical  determinant  of  familiar  type,  and  it  is  well  known  that 
all  its  roots  are  real.  For  the  system  of  the  eight  major  planets  it  is  of  the 
eighth  order.  It  is  most  unlikely  that  the  equation  would  have  exactly  equal 
roots  in  a  case  like  this,  nor  does  it  in  fact  happen.  But  it  is  to  be  observed 
that  the  occurrence  of  repeated  roots  would  alter  in  no  way  the  essential 
circumstances.  The  main  point  is  that  the  definite  quadratic  form  can  always 
be  reduced  to  the  form  So-ipf  by  a  linear  transformation  to  normal  coordinates. 
The  effect  of  repeated  roots  can  be  seen  when  there  are  three  planets.  Then 
SojjSi2  corresponds  to  an  ellipsoid,  which  is  one  of  revolution  when  two  roots 
at  are  equal.  An  arbitrary  element  enters  into  the  direction  cosines  of  the 
principal  axes,  which  are  the  coefficients  of  the  transformation.  But  this  does 
not  affect  the  form  of  the  result  or  the  stability  of  the  motion.  It  is  not 
necessary  to  examine  the  algebra  of  the  subject  further,  but  so  much  should 
be  mentioned  because  from  the  time  of  Lagrange  to  Weierstrass  in  1858  it 
was  supposed  that  the  occurrence  of  repeated  roots  would  result  in  the 
appearance  of  the  time  outside  the  periodic  functions  and  would  be  fatal  to 
stability.  It  is  not  so. 

180.  It  has  been  seen  that  the  orthogonal  transformation  to  normal 
coordinates  is  also  canonical  and  that  the  principal  function,  as  far  as  the 
eccentric  variables  are  concerned,  takes  the  form 


where  crf   is  positive,  since   V  is  a  positive  definite  form.     The   canonical 
equations  therefore  become 

dp{  dqi 

~di  =  "iqi>      dt=  -** 
and  the  solution  is 

Pi  =  d  cos  ((tit  +  h^,     qi  =  -Ci  sin  (o^  +  hi) 
where  Ciy  hi  are  arbitrary  constants.     This  gives  the  quadratic  integrals 

<n.2_L  tf.2  —  f7.2 

PI   <  yi  -  ^i  - 

These  results  are  immediately  expressed  in  terms  of  the  previous  variables 
&,  in.     Thus 

&  =  2  dit  j  PJ  =     2  ait  j  Cj  cos  (<Xjt  +  hj) 

i)i  =  2a»,  j  q.j  =  —  ZciiC  sin  («,-£  -I-  h) 


179,  iso]  Secular  Perturbations  201 

where  ait  j  are  definite  constants.     When  the  transformation  is  reversed, 


=     ait  j    ,     qj  =     ai 


and  the  quadratic  integrals  become 


The  general  solution  may  also  be  written,  with  the  degree  of  approxima- 
tion adopted, 

6iL^  cos  is  i  =  ^aitjGj  cos  (ttjt  +  hj) 
j 

6iL^  sin  CT{  =  2at-  j(7j  sin  («,-£  +  ^-) 
.; 

which  determine  the  eccentricities  and  the  motions  of  the  perihelia.  The 
question  then  arises  in  every  case  :  has  the  perihelion  a  mean  motion  ?  In 
other  words,  is  the  motion  of  perihelion,  to  use  the  analogy  of  the  simple 
pendulum,  of  the  circulating  or  the  oscillating  type  ? 

The  problem,  stated  in  general  terms,  is  not  a  simple  one.  But  there  is 
one  simple  case  which  will  serve  to  explain  what  is  meant  and  the  necessary 
condition  of  which  is  satisfied  more  often  than  not.  The  preceding  equations 
may  be  regarded  as  applying  to  certain  coplanar  vectors  whose  tensors  are 
6iL^,  ciijCj.  From  this  point  of  view  the  one  vector  is  represented  as  the 
sum  of  a  set  of  vectors  each  rotating  uniformly.  Let  the  tensor  of  one  vector 
of  the  set  exceed  in  length  the  sum  of  the  tensors  of  the  rest,  and  let  this 
vector  terminate  at  the  origin,  the  others  forming  a  chain  from  the  other 
end.  It  is  then  geometrically  obvious  that  the  representative  point  at  the 
end  of  the  chain  must  share  in  the  circulation  round  the  origin  of  the  pre- 
dominant vector.  The  perihelion  in  this  case  has  a  mean  motion  therefore, 
and  it  coincides  with  that,  ai}  associated  with  the  large  coefficient.  The 
sense  of  this  mean  motion  is  always  direct,  since  a^  is  positive.  In  the  same 
circumstances  ei  cannot  vanish,  but  has  a  lower  positive  limit. 

The  condition  is  clearly  satisfied  when  there  are  only  two  planets,  unless 
the  two  tensors  are  equal.  In  this  exceptional  case  it  is  evident  that  the 
mean  motion  of  a  perihelion  is  the  same  as  that  of  the  resultant  of  the  two 
vectors  and  is  the  arithmetic  mean,  \  (cr2  +  «3),  between  their  angular  motions. 

The  eight  roots  of  the  fundamental  determinant  range  between  the  values 
0"'616  and  22"*46  (Stockwell).  These  are  annual  motions,  so  that  the  corre- 
sponding periods  lie  between  58,000  and  2,100,000  years.  Since  they  are  of 
this  order  it  is  evident  that  ei?  tn-t-  can  be  developed  in  powers  of  the  time  and 
that  the  lowest  terms  of  such  expressions  will  suffice  to  represent  the  changes 
for  several  centuries.  These  are  the  secular  inequalities  as  commonly  under- 
stood, and  it  will  be  seen  that  they  exhibit  the  initial  changes,  apart  from 
those  of  short  period,  rather  than  truly  secular  effects. 


202  Secular  Perturbations  [CH.  xvi 

181.  These  results  for  the  eccentricities  and  perihelia  apply  almost 
without  change  equally  to  the  inclinations  and  nodes.  But  there  are  two 
differences  to  be  noted.  In  the  first  place  the  principal  function  is  a  negative 
definite  form,  which  may  be  written  after  the  transformation  to  normal 
coordinates, 


where  &•  is  positive.  In  the  second  place,  one  fit  is  zero,  or,  in  other  words, 
the  discriminant  or  Hessian  of  V  (a  quadratic  form)  vanishes.  For  the  part 
which  involves  the  oblique  variable  £f  may  be  written  (§  178) 


and  therefore 


If  then  i  is  the  characteristic  of  a  row  and  j  of  a  column  in  the  Hessian,  and 
each  column  is  multiplied  by  the  corresponding  Lj  ,  the  sum  of  each  row  will 
vanish.  Hence  the  discriminant  is  identically  zero  and  ft  —  0  is  a  root  of  the 
fundamental  equation. 

The  physical  reason  for  this  is  easily  seen.     For  the  canonical  equations 
become 


dt~ 
Corresponding  to  the  root  fti  =  0, 

Pi  =  26f,j  £j  =  const.,     qi  =  S&jj  rjj  =  const. 

which  are  two  linear  integrals.  The  constants  need  not  be  zero,  and  the 
inclinations  may  be  finite,  while  their  variations  vanish.  This  in  fact  is  the 
case  when  the  orbits  are  all  coplanar  and  inclined  to  the  plane  of  reference. 
This  explains  why  the  fundamental  determinant  has  a  zero  root.  The  other 
seven  negative  roots  when  calculated  for  the  solar  system  are  quite  similar  in 
magnitude  to  the  positive  roots  of  the  determinant  in  a. 

The  general  solution  of  the  equations  when  a  finite  root  is  in  question  is 

Pi  =  Di  cos  (fat  +  ki),     qt  =  D{  sin  (fat  +  k{) 
giving  the  quadratic  integrals 


3  3 

From  the  general  solution  it  follows  that 

iilf  cos  Hi  =  &  =  ^bi,jpj  =  ^bijDj  cos 
-  iiLi  sin  £li  =  i]i  =  2&i<   =  26;  D  sin 


181,  IBS]  Secular  Perturbations  203 

where  bij  are  the  definite  constants  of  the  transformation  to  normal  coordi- 
nates. Owing  to  the  zero  root  in  ft,  t  disappears  from  one  term  on  the  right- 
hand  side  of  each  equation,  leaving  seven  periodic  terms  and  one  constant, 
but  the  form  is  undisturbed  by  this  fact. 

These  equations  determine  the  inclinations  and  the  motions  of  the  nodes. 
The  plane  of  reference  is  fixed  and  arbitrary,  except  in  so  far  as  it  lies  near 
the  average  plane  of  the  orbits.  Considered  as  applying  to.  ajset  of  rotating 
coplanar  vectors,  the  equations  show  immediately  that  if  one  coefficient  on 
the  right  exceeds  the  sum  of  all  the  rest  (taken  positively),  the  node  has  a 
mean  motion  equal  and  opposite  to  that  of  the  corresponding  vector,  and  this 
mean  motion  is  therefore  retrograde.  When  this  simple  criterion  is  satisfied, 
as  it  is  more  often  than  not,  it  is  also  evident  that  the  tensor  of  the  vector 
iiLf  cannot  vanish  and  that  i{  has  a  lower  limit. 

182.     The  sum  of  the  quadratic  integrals  gives 

2  (pr  +  qi2}  =  2  (&»  +  V)  =  const. 

and  this  applies  separately  to  the  eccentric  and  to  the  oblique  variables.  It 
follows  immediately  from  the  canonical  equations  of  §  179  without  any  trans- 
formation. Now  %i,  rji  contain  the  factor  Li,  which  is  mi  (ml  +  TOf)*,/*^-^""1^* 
or  to  the  lowest  order  in  the  masses  mim^a?.  Hence 


?  =  const. 

iObii?  =  const. 
or,  as  the  latter  is  more  usually  written, 

STO^CI^  tan2  it  =  const. 

for  the  degree  of  approximation  adopted  allows  of  no  discrimination  between 
these  forms.  The  constants  being  small  initially  it  follows  that  the  orbit 
of  no  considerable  mass  in  the  system  can  acquire  an  indefinitely  large 
eccentricity  or  inclination  at  the  expense  of  the  others  as  a  result  of  mutual 
perturbations.  These  propositions,  due  to  Laplace,  clearly  have  an  importance 
analogous  to  that  of  Poisson  on  the  invariability  of  the  mean  distances. 

The  areal  velocity  in  any  orbit  is 

(/j.p)^  =  (TO!  +  wii)*0i*  cos  fa  =  G{. 

The  mass  factors  being  TO^^i/^"1  as  in  §  170,  the  components  of  angular 
momentum  are 

(ziTOi/if-!/^"1  (sin  ii  sin  H;,  -  sin  it  cos  ftt-,  cos  ii) 

=  Li  cos  fa  (sin  ii  sin  H;,  —  sin  i{  cos  fl^,  cos  ii) 

when  the  direction  cosines  of  the  normal  to  the  orbit  are  introduced.  These 
components  may  be  written  (§  174) 

—  77^  2  Lf  cos*  fa  cos  ^  ii  ,    —  git  2  Lf  cos^  fa  cos  ^  it  ,     Li  cos  fa  cos  it 


204  Secular  Perturbations  [CH.  xvi 

or  since 

f2*M  +  *ft  i  =  %Li  (1  -  cos  &),     f  \2  +  r,\z  =  2£;  cos  0<  (1  -  cos  i,-) 

they  can  also  be  expressed  in  terms  of  these  quantities.  The  areal  integrals 
then  become 

-  Sift,  {Lt  -  ±  (f  f>1  +  rf\  ,)  -  i  (£\2  +  7ft-  2)|*  =  const. 

-  2ft,,  (Z,  -  $  (f  V  +  ift,)  -  1  (r,)2  +  7?\2)}*  =  const. 

S  \Li  -  %  (f  V  +  ift  ,)  -  £  (pi>2  -f  77\2)}  =  const. 

If  the  plane  01  reference  is  the  invariable  plane  the  first  two  of  these  con- 
stants are  zero.  In  that  case,  when  there  are  only  two  planets,  f)2/J;2  is  the 
same  for  both  and  the  nodes  coincide,  which  is  the  property  already  noticed 
in  §  169  and  referred  to  as  the  elimination  of  the  nodes. 

These  integrals,  being  satisfied  identically,  remain  true  when  developed 
according  to  order  and  rank.     Thus  the  third  equation  gives 


+  ift!  +  pil2  +  <n\2)  =  const. 

which  is  the  sum  of  the  quadratic  integrals  both  for  the  eccentric  and  the 
oblique  variables.  For  Lt  has  no  terms  of  zero  rank,  and  the  purely  periodic 
terms  of  the  first  order  are  excluded  from  consideration. 

Thus  Li  is  for  the  present  purpose  to  be  regarded  as  constant.  The 
neglect  of  terms  of  the  fourth  degree  in  the  disturbing  function  implies  the 
neglect  of  the  third  degree  in  the  variables  £,  77  themselves.  Hence  to  the 
same  approximation  the  first  two  areal  integrals  give 

2£<"ift8  =  const.,     ££{*£{,,  =  const. 

These  then  are  the  two  linear  integrals  found  above  for  the  oblique  variables, 
and  their  physical  meaning  is  thus  explained.  The  constants  are  now 
interpreted  (to  a  factor)  as  the  angular  momenta  of  the  system  about  two 
rectangular  axes  in  the  arbitrary  plane  of  reference.  In  particular,  if  the 
invariable  plane  of  the  system  is  taken  as  the  plane  of  reference,  both  the 
constants  will  become  zero. 

183.     The  interpretation  of  the  equations 


cos  v        ~  cos 

.     or*  =  zoiiCj   . 
sm  J  sm 


in  a  vectorial  sense  has  been  seen  to  give  a  lower  limit  of  €i  when  one  of  the 
tensors  |  a^jGj  exceeds  the  sum  of  the  rest.  In  all  cases  similar  reasoning 
shows  that 

et-Z//<  2 


182,  iss]  Secular  Perturbations  205 

gives  an  upper  limit  of  the  eccentricity.     Similarly  the  inequality 


gives  an  upper  limit  of  the  inclination.  The  actual  limits  found  in  this  way 
by  Stockwell  are  of  interest  and  are  therefore  reproduced. 

Eccentricity  Inclination 

Max.  Min.  Max.  Min. 

Mercury         O232  0121  9°'2  4°'7 

Venus            0-071  ...  3'3 

Earth             0'068  ...  31 

Mars               0140  O'OIS  5'9 

Jupiter           0-061  0'025  0'5  0'2 

Saturn           0'084  0'012  I'O  0'8 

Uranus          0'078  0'012  11  0'9 

Neptune        0'015  O'OOG  0'8  0'6 

The  effect  of  periodic  inequalities  is  ignored,  and  the  inclinations  are  referred 
to  the  invariable  plane.  Minimum  figures  are  given  only  when  a  pre- 
ponderating term  exists. 

Since  Lf  contains  mf  as  a  factor  these  limits  have  no  value  when  the 
mass  mi  is  very  small.  To  consider  this  case  let  an  infinitesimal  mass  m0  be 
added  to  the  system.  Then  for  the  eccentric  variables, 


Inspection  of  the  explicit  form  in  §  178  shows'  that  A{j  is  of  the  order  of  rat-, 
any  of  the  masses,  assumed  comparable,  of  the  finite  planets ;  that  A0j  is  of 
the  order  of  mfnif ;  and  that  A0>0  is  again  of  the  order  mi. 

The  canonical  equations  give  for  the  infinitesimal  planet 
d&=     A 

— -  =  -A    £  -2A   •£• 

As  the  new  mass  is  regarded  as  infinitesimal,  the  motion  of  the  finite  planets 
will  not  be  influenced,  and  the  former  solution 

fy  =     So,-, i Ci  cos  (ait  +  hi) 

tjj  =  —  ^a>j,iCi  sin  (att  +  h,) 
holds  good.     Hence 

-^  —  A0>0r)0  =  —  ^Aoja^tCi  sin  (a^  +  Af) 

dfln 

-3;  +  At  0£0  =  —  2  A0  jdj  iCi  cos  (dit  +  hi), 
dt  ij 


206  Secular  Perturbations  [CH.  xvi 

These  are  the  equations  for  a  natural  oscillation,  together  with  a  set  of  forced 
oscillations,  and  the  solution  is 


0  =     «0  cos      0,0*  +  0  -       ojdj^i       ,o  —  "t"1  cos 
«7o  =  —  «<>sm  040,o*  +  ^o)  +  2A0tjOjtiCi  (A0i0  -  a,-)-1  sin 

where  afl,  h0  are  arbitrary  constants.  In  general  this  solution  shows  that  the 
eccentricity  (and  a  similar  form  applies  to  the  inclination)  of  the  orbit  of  the 
infinitesimal  mass  will  remain  small.  For  £0,  »?o  contain  m^  as  a  factor,  and 
A0>j(A0t0  —  at;)"1  is  of  the  order  of  mfmt~~.  An  exception  occurs  when  A0>0 
is  nearly  equal  to  o^,  that  is,  when  the  period  of  the  free  oscillation  nearly 
agrees  with  one  of  the  forced  periods  imposed  by  the  main  planetary  system. 
The  corresponding  amplitude  then  tends  to  become  infinite.  This  condition 
is  fulfilled  at  the  mean  distance  from  the  Sun  1*95,  or  near  the  inner  limit  of 
the  minor  planets  (Eros  excepted),  but  for  the  inclinations  only  (A0j0  —  fit). 
But  before  any  positive  conclusion  can  be  drawn  for  this  case,  the  extremely 
limited  development  of  the  disturbing  function  must  be  remembered*. 

*  Cf.  Charlier's  Mechanik  des  Himmels,  i. 


CHAPTER    XVII 

SECULAR    INEQUALITIES.       METHOD    OF    GAUSS 

184.  A  beautiful  method  of  calculating  the  secular  perturbations  of  the 
first  order,  due  to  the  action  of  one  planet  on  another,  was  proposed  by  Gauss 
in  1818.  It  was  this  method  which  was  applied  by  Adams  to  the  path  of  the 
Leonid  meteors.  Further  developments  have  been  given  by  several  writers, 
and  references  will  be  found  in  an  article*  by  H.  v.  Zeipel. 

The  principle  of  the  method  is  extremely  simple.  Equations  for  the 
variations  of  the  elements  have  been  found  in  a  suitable  form  in  §  142.  As 
an  example  we  may  take  (/u.  =  n2as) 

di       1     r  W  cos  u 
dt     no?  '     cos  <j> 

Here  the  right-hand  side  can  be  developed  in  terms  of  M,  M',  the  mean 
anomalies  of  the  disturbed  and  disturbing  planets,  in  the  form 

di 

|  .  A^  +  1  AJJ  cos  (jM+j'M1  +  q) 

and  hence,  the  coefficients  being  constant  in  the  first  approximation, 

i  -  i,  =  A0>0  t  +  2Aj>f  sin  (jM+-j'M'  +  q)/(jn  +j'ri). 

If  therefore  the  mean  motions  n,  n'  are  incommensurable,  so  that  (jn  +j'ri) 
can  never  vanish,  AQi0t  constitutes  the  secular  inequality  in  i.     Now 

VfJi~\  1       f27r   f  2ff  di 

l*fr  -L  U/l      711,f-     7Ti,r/ 

-v-  a   =4^   J  dtmm 

|_CfcCJo,  0          ^"      JO      JO       *f* 

1  f27r  F  1      f2ir  ~l 

—        rcos  J     -        WdM'ldM  ...(1) 
2-Trna2  cos  <f>  .'  0  \^TT  J  0 

The  component  W  contains  as  a  factor  &2m'  =  n2a3m'/(l  +  m).     We  therefore 
write 


+m 


with  similar  reduced  mean  values  S0,  T0  corresponding  to  S,  T.     If  then  a 
series  of  values  of  S0,   T0>  WQ  can  be  calculated  for   a  number  of  points 

*  Encyklopddie  d.  math.  Wiss.,  vi.  2,  p.  632. 


208  Secular  Inequalities  [CH.  xvii 

regularly  distributed  round  the  disturbed  orbit,  they  can  be  introduced  into 
the  equations  for  the  variations  and  a  simple  quadrature  will  give  the 
secular  perturbations  of  the  several  elements,  that  of  a  being  zero. 

185.  In  calculating  S0,  T0,  W0,  the  disturbed  planet  occupies  a  given 
fixed  point  P  in  its  orbit.  It  is  clear  that  S0,  T0,  W0  are  components  of  the 
mean  attraction,  with  respect  to  the  time,  exercised  at  P  by  a  unit  mass 
describing  the  disturbing  orbit,  with  unit  constant  of  gravitation.  They  are 
the  same  as  would  result  if  the  disturbing  orbit  were  permanently  loaded  so 
as  to  constitute  a  material  ring  of  the  same  total  mass,  when  the  density  is 
proportional  to  dM'  Ids'.  Thus  it  is  necessary  to  calculate  the  attraction  of 
an  elliptic  ring  of  this  kind. 

Let  any  system  of  rectangular  axes  xyz  be  taken,  with  origin  at  P.  Let 
(x0,  y0,  z0)  be  the  coordinates  of  the  Sun,  (x',  y,  z')  those  of  a  point  P'  on  the 
disturbing  orbit,  and  let  da  be  the  area  of  an  elementary  focal  sector,  dV 
the  volume  of  the  tetrahedron  on  the  base  da'  with  its  apex  at  P.  Then 

2p .  da-'  =  6d  V  =  x6  (y'dz  -  z'dy')  +  y0  (z'dx  -  x'dz)  +  z0  (x'dy  -  y'dx) 

where  p  is  the  perpendicular  from  P  on  the  plane  of  da'.  Hence  one 
component  of  the  required  attraction  at  P  is 

1 


Px      27rJo  A3 

where  a',  b'  are  the  semi-axes  of  the  disturbing  orbit  and  A2  =  x'z  -f  y'2  +  z'*. 
This  takes  account  of  the  first  (principal)  part  of  the  disturbing  function 
only:  the  second  (indirect)  part  has  been  left  out  of  consideration  because 
(§  159)  it  gives  rise  to  no  secular  terms  in  the  perturbations  of  the  first  order. 
It  is  now  to  be  observed  that  x'&~sdV  is  a  homogeneous  function  of  degree  0 
in  x',  y',  z',  and  can  therefore  be  expressed,  since  z'dy'  —  y'dz'  —  z'2d  (y'/zf), . . . , 
in  terms  of  x'/z,  y'/z',  which  are  connected  by  some  relation 

f(x'/z',    3/7*')  =  0 

which  is  the  equation  of  the  cone  having  its  apex  at  P  and  the  attracting 
ring  as  its  section.  Thus  the  integral  factor  of  Px  (and  similarly  of  Py,  Pz) 
depends  only  on  the  form  of  the  cone  and  not  on  the  particular  section. 
This  is  true  whatever  the  shape  of  the  ring  may  be.  But  in  the  present  case 
the  cone  is  of  the  second  degree,  and  the  axes  may  now  be  identified  with 
its  principal  axes,  P  (X,  Y,  Z).  Let  PZ  be  the  internal  axis  and  a,  ft  the 
semi-axes  of  the  section  Z=l.  The  coordinates  of  P'  can  be  written 


Y'  =  /3sinT,     Z'=l 
where  r  is  the  eccentric  angle  in  the  section,  and 

A2  =  1  +  a2  cos2  T  +  /32  sin2  T,     6d  V  =  (-  ftX,  cos  T  -  a  F0  sin  r  +  aftZ,)  dr. 


184-186]  Method  of  Gauss  209 

Hence 

p    = 1__  [2w  «  cos  T  (-  ffZ0  cos  r  -  « FQ  sin  T  +  afiZ,)  dr 

Zjra'b'p  J0  (i+a*  COS2  T  +  £»  Sin2  T)f 


-  2«/3A"0  f  *»•  cos2  T  i 

~/o    "A^ 


and  similarly 

P    •     ~  2gffF°  ( ^  sin2TC?T       p    =  2a^,  r*"  dr 
Tra'&'p    J0         A3  z~Tra'b'p]Q    A5' 

These  components  can  now  be  expressed  in  terms  of  the  complete  elliptic 
integrals 

HTvd-l-sin-T)'     *-£  *1-»«*T>*. 

For,  since 

sin  T  cos  r          cos2  r  —  sin2  T  +  &2  sin4  T 


**       sin2rc?T        ,!„  1          ,, 

1     "T"      ln-^  Tn/-<  *1«\      -C'* 


Hence 


(«2-/32)  V(l  +  a2) 
•na'b'p '  (a2  —  /32)  V(l  +  fl2) 

OX  ~/3 

P*  = 


where  the  modulus  k  of  E  and  JP  is  given  by 

«2  _    /02  1      l      0-2 

^•2  —  —      "        1      P  —         p 

IV      —      -  _    .  ±    "~"    A/      _  ~T    • 

1  +  a2  '  1  +  a2 

186.  It  is  now  necessary  to  consider  the  geometry  of  the  problem.  Let 
the  angular  elements  of  the  disturbed  orbit  be  fl,  *',  «<>,  and  of  the  disturbing 
orbit  H',  i',  w'.  These  are  referred  to  the  ecliptic,  which  it  is  convenient  to 
eliminate  by  referring  the  latter  orbit  directly  to  the  former.  With  some 
change  in  the  notation  of  §  67  the  equations  there  found  give 

sin  ^  (ft"  +  &)'  -  w")  sin  |i"  =  sin  £  (11'  -  ft)  sin  \  (if  +  i) 
cos  |  (ft"  +  a)'  -  w")  sin  £tv/  =  cos  £  (ft'  -  ft)  sin  |  (iv  -  i) 
sin  £  (ft"  -  w'  +  &>")  cos  ^i"  =  sin  £  (ft'  -  ft)  cos  £  (i'  +  i) 

COS  ^  (ft"  -  0)'  +  ft)")  COS  £l'"  =  COS  |  (ft'  -  ft)  COS  |  (t'  -  l). 
P.  D.  A.  14 


210  Secular  Inequalities  [CH.  xvn 

Here  H"  is  the  distance  of  the  intersection  of  the  two  orbits  from  the  ecliptic 
node  of  the  disturbed  orbit,  i"  is  the  mutual  inclination  of  the  two  planes, 
and  to"  is  the  distance  of  the  perihelion  of  the  disturbing  orbit  from  the 
intersection. 

Two  sets  of  rectangular  axes,  with  an  arbitrary  origin  0,  are  now  to  be 
defined.  For  0  (£,  77,  £)  the  directions  are  those  of  S,  T,  W,  so  that  0%  is 
parallel  to  the  radius  vector  at  P,  Orj  is  parallel  to  the  plane  of  the  disturbed 
orbit  and  90°  in  advance  of  0%,  and  0%  is  in  the  direction  of  the  N.  pole  of 
this  orbit.  For  the  second  set,  0  (x,  y,  z),  Ox  is  directed  towards  the  peri- 
helion of  the  disturbing  planet,  Oy  is  parallel  to  the  plane  of  the  disturbing 
orbit  and  90°  in  advance  of  Ox,  and  Oz  is  directed  towards  the  N.  pole  of  this 
orbit.  Let  v  be  the  true  anomaly  at  P,  and 

to  +  v  —  fl"  =  v1 

the  distance  of  P  from  the  intersection  of  the  orbits.  Then  the  relations 
between  the  two  systems  of  coordinates  are  given  by  the  scheme : 

£  i}                                   (T 

a;      cos  to"  cos  #!+ sin  to  "  sin  Vi  cosi"  —  cos  to"  sin  Vj+ sin  to"  cos  vl  cos  i"  sino/'sint" 

y  —  sin  to" cos #!  + cos  to" sin #! cosi"  sin  to"sinv1  +  costo"cosy1cost"  cos  w"  sin  t' 

z                         —  sin?^ sin  i"  —  cosv^sini"                        cosi" 

Thus  if  r  is  the  radius  vector  at  P,  and  the  origin  0  be  taken  at  the  centre 
of  the  disturbing  orbit,  the  coordinates  of  P  are  (xlt  y1}  z^,  where 

^  =  ae  +  r  (cos  to"  cos  vl  +  sin  to"  sin  vl  cos  i") 
yi  =  r(—  sin  to"  cos  v±  +  cos  to"  sin  vt  cos  i"),     zl  =  —  r  sin  v1  sin  i"  =  p 
and  a,  e  are  the  mean  distance  and  eccentricity  of  the  disturbing  orbit. 

187.  Consider  now  the  con  focal  system  of  quadrics  of  which  the 
disturbing  orbit  is  the  focal  ellipse 

&*&rlm 

The  parameters  \,,  X2,  X3  of  the  three  quadrics  passing  through  the  point 
(xi>  y\  >  zi)  are  given  by 

/».  2  „.  2  „  2 

•^      i     3fr      if?  _i 
a'2  +  X     b'-  +  X      X 

or  as  the  roots  of  the  cubic 

X3  -  X2  (x?  +  y?  +  z?  -a'*-  b'2) 

+  X  (tt'2&'2  —  a'2;z/i2  —  fc^a?!2  —  a'-Z]2  —  6'2^j2)  —  af*V*z*  =  0   (2) 

Now  the  axes  of  any  tangent  cone  to  a  quadric  are  the  normals  to  the  three 
confocals  which  can  be  drawn  through  the  vertex  of  the  cone,  and  this 
remains  true  in  the  particular  case  where  the  focal  ellipse  is  a  section  of 


186,  is?]  Method  of  Gauss  211 

the  cone.     Hence  the  relations  between  the  sets  of  coordinates  (  X,  Y,  Z)  and 
(#,  y,  z]  are  given  by  the  scheme  : 

x  y  z 

X 


Y        #,«!  (a'2  +  X^-1         p,yi(b'2  +  \2)~1 
Z        p3x,  (a2  +  A3)-1         psft  (I)'1  +  \3)~l 

where  pl;  p2,  ps  are  such  that 

Pl*  (*,«  (a2  +  X,)-8  +  y?  (6'2 


When  combined  with  the  scheme  given  above  for  (x,  y,  z),  (f,  77,  £),  this  gives 
the  relations  between  (X,  Y,  Z)  and  (£,  77,  £). 

The  equation  of  the  cone  is 

(ex,.  -  xz$     (zyl  -  yz,)- 

0'»-  —j/2—  '*$ 

for  this  is"  clearly  homogeneous  and  of  the  second  degree  in  as  —  a;l,y  —  yl, 
z  —  Z-L  ,  and  its  section  by  the  plane  z  =  0  is  the  disturbing  orbit.  Transposed 
to  parallel  axes  through  its  vertex  (xl}  ft,  z^)  it  becomes 


____ 

'2       '2        *      '2       /a 


___ 
a'2     6'2     z*      '2     6/a        /       &''  '  ^  "    a'2  ' 


The  justification  for  identifying  these  two  forms  is  seen  on  comparing  the 
three  functions  of  the  coefficients  which  remain  invariant  under  a  rotation 
of  the  axes.  It  will  then  be  found  that  the  results  are  equivalent  to  the 
relations  between  the  coefficients  and  roots  of  (2). 

It  is  convenient  to  write  down  the  equation  of  the  reciprocal  cone.  The 
coefficients  are  the  minors  of  the  discriminant  of  the  previous  equation 
^_!  =  0.  Hence  with  due  care  in  choosing  the  right  multiplier  the  desired 
equation  may  be  written 

x2  (acf  —  a'2)  +  y'2  (y^  —  6'2)  +  z'*z?  +  2yz  y^  +  i2.zxzlxl  +  ^xyx^ 
=  XaZ2  +  X2F2  +  X3£2  =  Fl  =  0 

the  invariant  relations  being  identical  with  those  between  the  coefficients 
and  roots  of  (2). 

Also 

&  +      *  +  #  =  X*  +    F2 


and  it  is  evident  that  F_lt  Fl  can  also  be  readily  expressed,  by  means  of  the 
transformation  scheme  of  §  186,  in  terms  of  £,  rj,  £. 

14—2 


212  Secular  Inequalities  [CH.  xvn 

188.  Two  of  the  roots  of  the  cubic  (2)  are  negative  and  one  positive, 
since  two  of  the  corresponding  quadrics  are  hyperboloids  and  one  an  ellipsoid. 
Let 

Xj  <    X2  <  0  <    A3. 

The  axis  of  Z  is  then  the  internal  axis  of  the  cone  F^  and  it  follows  that 


X3 '  X3 '  1  +  a2      X3  —  Xt 

The  elliptic  integrals  F,  E  can  therefore  be  found.  The  coordinates  of  the 
Sun  relative  to  the  point  P  are  x0  =  a'e'  —  x1,  y0  =  —  yl}  z0  =  —  z1  in  the  system 
(x,  y,  z)  and  (X0,  Y0,  Z0)  can  be  deduced  by  the  transformation  scheme  of 
§  187.  Hence  PX,  PY>  PZ  become  known,  and  the  components  Pf  =  S0> 
Pn  =  T0,  Pf  =  W0  may  be  derived  by  applying  the  two  transformations  of 
§§  186  and  187. 

It  is  unnecessary  here  to  consider  the  equations  for  all  the  inequalities. 
As  a  type,  (1)  now  becomes 

'di\  nam  1    /"2ir  Tir  , ., 

r       =  -  T  •  5—        ^  cos  u .  W0dM. 

\dt/0to      (1  -f  m)  cos  </>    2?!-  J  0 

Suppose  that  j  values  i/rg  of  ty  =  r  cos  u  .  W0  have  been  found,  corresponding 
to  j  points  around  the  disturbed  orbit  at  which  M  has  equidistant  values, 
0,  2-TT/j, . . . ,  2  (j  -  1 )  if  I  j-  Then  (Chapter  XXIV) 

T|T  =  a0  +  2  di  cos  i M  +  S  bi  sin  iM 
where 

a0  =  -  S^g,     «{ =  -  S^g  cos  — -. — ,     b{  =  -  2 ^g  sin  — ; —  . 
J  *  J  s  J  J  s  J 

Hence 

nam' 

•o    (3) 


(-}     = 

\dtJ0>0 


(1  +  m)  cos  <f) ' 

and  it  is  only  necessary  to  calculate  the  average  value  of  i|rg  to  have  the 
secular  inequality.  For  the  major  planets  j—  12  practically  suffices.  The 
summation  formula  for  a0  really  gives  a0  +  «j  +  ... .  It  is  therefore  necessary 
to  take  j  large  enough  to  make  a.j  negligible.  The  number  of  points  to  be 
taken  on  the  disturbed  orbit  thus  depends  on  the  practical  convergency  of 
the  series  a0,  a1}  a2,  — 

It  is,  however,  preferred  to  take  points  equidistant  in  E,  the  eccentric 
anomaly,  instead  of  M,  since  this  secures  a  more  even  distribution  in  arc. 
The  advantage  of  this  course  seems  scarcely  obvious,  because  it  appears  to 
weight  unduly  the  part  of  the  orbit  which  is  passed  over  rapidly.  But  the 
modification  is  easily  made.  In  this  case 

•\Jr  =  a0  +  2at-  cos  iE  +  £&t-  sin  iE 


188,  189]  Method  of  Gauss  213 

where  again 


1  ^  ,  2  „  ,          2*wr       .       2  _,    .    2s 

a0  =  -  2,  y-g,     af  =  -  2,  YS  cos  —  —  ,     bi  =  -  2,  sin  —  s 

J    s  ^s  J  J    s  J 

but  the  meaning  of  ^  will  be  changed.     For 

dM  =  (l-e  cos  #)  dE  =  a^r  .  d# 
and  (1)  may  be  written 

fdi  •  nam  1    f  2 


w  ,  „ 
a-1*-8  cos  u  .  W0dE. 

Hence  (3)  will  still  hold  good  if  a0  is  the  simple  mean  value  of  -fy,  where  i/r 
is  now  a-1r2  cos  w  .  W0. 

189.  The  cubic  (2)  has  three  real  roots  and  can  be  easily  solved.  It  is 
now  to  be  seen  that  the  solution  can  be  avoided.  Let  the  equation  be 
written 

\3  -f  3&jX2  +  3&a\  +  ks  =  0 

the  roots  being  X1?  Xj,  X3,  and  let  the  result  of  removing  the  second  term  be 

4X*-0bX'-'9,-0 

of  which  the  roots  are  e1}  e.2)  es.     Then 

#2  =  -  4  (e.2e3  +  esei  +  ^62)  =  12  (fcj2  -  &,) 


and 

3^  =  2X1-  \s-\3,     3e2=2\a-Xs-\1,     3es  =  2X3  -  \  -  \2 

e1<e2<es,     e1  +  e2  +  e:i  =  0. 
Thus 

A2  =  1  +  a2  cos2  T  +  /32  sin2  T  =  XT1  {(X;!  -  Xj)  cos2  r  +  (X8  -  X,,)  sin2  r} 

=  ^-r1  {(e3  ~  «i)  cos2  r  +  (ea  -  e2)  sin2  r}  =  Xs"1  A' 
and  the  components  to  be  calculated  are 

-2Z0(X1XaX3)*  /•*"•  cos2  rdr      p        -2F0(X1X2X3)^  r*»  sin2T^r 
7ra'6>          Jo        A'3  ira'b'p     ~]0    ~A^~  ' 


where  X1X2X3  =  —  7<;:).     It  is  clearly  possible  to  write  consistently 


whence 

dr      (e3  —  e-,)  (e2  —  e2) 
2  sin  T  cos  T  -7-  =  )-^ 

as      (e2  —  61)  (s  —  es)2 

and 


-et)  («-««)(*-«;) 


214  Secular  Inequalities  [OH.  xvn 

But  this  can  be  written 

A'"1  dr  •=  dv,     $>(u)  =  s 

where  $(u)  is  the  Weierstrassian  elliptic  function  formed  with  the  roots 
BI,  ez,  e3.  When  r  =  0,  $(u)  =  e2,  w  =  o>2;  when  T=^TT,  ^>(u)  =  el,  U  =  CDI. 
Hence 

d-r      /•«"        >  M  ~  e3       d    = 

^-^-u,,       *» 

,    _  ~     £(u)  +  e2u     ""«  _        77  4- 

" 


Jo        A"          ].fe<*-4)(4:- 

Jo        A/:1      =  L,  (e2  -  6l)  (e,  -  ej  du  =  |_(e2  -  e,)  (ea  -  eO_L  =  02  -  *i)  (es  -  ej 
where 

The  quantities  &>  and  rj  will  now  be  found. 

190.  The  reader  who  is  unacquainted  with  the  theory  of  elliptic  functions 
will  notice  that  nothing  beyond  the  definitions  of  the  functions  jjp(w),  £(w)  is 
here  involved,  and  that  these  can  be  easily  inferred.  In  fact,  if  the  variable  s 
be  retained,  it  is  easily  seen  that 

ds  [e*  sds 


/  [4>  (s  -  e,)  (s  -  e2)  (s  -  es)} 
where 

4  (s  -  e^  (s  —  e2)  (s  -  es)  =  4>s3  -  g2s  -  ga,     el<ez<e3. 

The  range  of  integration  is  the  finite  interval  between  the  roots  in  which  the 
integrals  are  real.     Let 

s  =  (i?2)4  cos  0,    cos  37  =  (Wg3*g.rrf  =g~*. 
The  values  of  6  corresponding  to  et,  e2,  es  in  order  are  clearly 

0]  =  I1""  +  7>     #2  =  ITT  -  7,     03  =  y<  ^TT 
since 

-  #2s  -  #3  =  (|0,)*  (cos  30  -  cos  87). 


Hence 

Yl    ^-*  <"*'  shifldfl  j/'*'         sin20c 

J  ^  V  (cos  30  -  cos  37)  '     ^  ~         (  M  J  e,  Vjcos  30  -  cos  37)  ' 
Now  the  Mehler-Dirichlet  integral*  gives 


P  (      i  \  -  1 
~7r 

where  Pn  denotes  Legendre's  function  of  the  first  kind  and  order  n.     Let 
</>  =  30  —  2-rr,  and  then 

/•«,          &(»+H*d0 

-_  J^2  7T6  ^D  -  Pn  (COS  37) 

J  e,  V  (cos  30  -  cos  87) 
*  Cf.  Whittaker's  Modern  Analysis,  p.  219;  Whittaker  and  Watson,  p.  308. 


189-191]  Method  of  Gauss  215 

whence 


Now  put  n  =  —  \  and  +  £  in  succession.     Thus 

-i 
"~  -(C       ^ 


/•«•          sm29d0  _i 

-77—    ^rx-~       -fo-q  =  -  6      *  7T  Pt  (COS  87). 

./  02  V  (cos  80  —  cos  87)  F  v 

But  the  Legendre's  functions  can  be  expressed  in  the  form  of  hypergeometric 
series*  F(—n,n  +  l,l,  sin2  §7).     Hence  finally 

,£,  I,sin"t7) 


where  sin2  f  7  =  £  (1  —  <7     ).      Thus  o>  and  77  are  expressed  in  a  form  not 
requiring  the  solution  of  the  cubic  equation. 

These  hypergeometric  series  are  not  the  same  as  those  originally  found 
by  H.  Bruns  as  the  solution  of  the  problem.  But  the  latter  are  easily 
deduced.  For  Pn(z)  satisfies  the  differential  equation 


The  result  of  changing  the  independent  variable  to  x=  1  —z2.is 


which  is  satisfied  by  the  hypergeometric  series  F(—^n,  ^n  +  ^,  1,  x).  When 
z  =  cos  87,  x  =  sin2  87  =  g~l  (g  —  1)  and  since  there  can  be  only  one  convergent 
series  for  y  in  powers  of  x,  this  is  it.  The  above  series  may  therefore  be 
replaced  by 

F  (TV,  f¥  1,  sin2  37),     F(-  iV  &>  1,  sin2  87) 
which  are  the  series  obtained  by  Bruns. 

191.  Let  the  origin  of  coordinates  now  be  taken  at  the  Sun,  the  point  P 
being  at  (X,  Y,  Z)  or  (-  X0,  -  Y0,  -  Z0).  Then  the  components  PX)  PY,  PZ 
(4)  can  be  derived  by  partial  differentiation  from  the  potential 


=  (-k)*[* 

Tra'b'p  Jo 


**  Z2  cos2  T  +  F2  sin2  r-Zzdr 


a'b'pJo  A'3 


•rra'b'p  '  (e,  -  e2)  (e,  -  BI)  (e2  -  ej 

Of.  Whittaker's  Modern  Analysis,  p.  214  ;  Whittaker  and  Watson,  p.  305. 


216 


where 


Secular  Inequalities 


#1  =  (e3  -  e2)  X2  +  (e,  -  e3)  F2  +  (e2-  e,)  Z* 
G2  =  e1  (es  -  e2)  X2  +  e2  (e,  -  e3)  Y2  +  e3  (ez  -  e,)  Z2 
Now  by  ordinary  multiplication  of  determinants 


and 


X2  Y2  Z2 

A.J  A2  /Vg 

111 

Z2  F2  Z2 

"X  — 1  "X  — 1  "X  —  1 

*M  'Vj  ^3 

1  1  1 


where 


111 

Li         Xa""1      A3—1 

\l  Xjj  X3 

1      1      1 


FQ 


2V 


[CH.  xvn 


and  e1}  e2,  ea  are  the   roots  in   X'   corresponding   to  X,,  X2,  X3.     The   first 
determinant  is  clearly  —  Gj,  and  the  determinant  below  it  is 

The  multiplying  determinant  in  both  identities  is 

—  (XJX2X5)""1  (X3  —  Xj)  (X3  —  X^)  (X2  —  Xj)  =  \KZ~I  (g<?  —  27flf32)" 

and  the  determinants  on  the  right-hand  side  are  easily  expressed  in  terms  of 
h,  k2,  k3.     They  are  respectively  9k3~lH1  and  —  9k3~2H2,  where 


and 


Hence 


'•-hk^  +  Fo 

144  (-  jfc,)* 


_,  (k,k2  - k3)  ks. 


.(5) 


ira'b'p  . '        g/ 

But  FI,  F0,  jP_!  have  been  expressed  (§  187)  in  terms  of  (as,  y,  z).  Hence  the 
system  of  coordinates  (X,  Y,  Z)  has  been  completely  eliminated  from  the 
problem. 

192.  Now  F  is  a  homogeneous  quadratic  function  in  (#,  y,  z)  and  can  be 
reduced  to  the  same  form  in  (£,  77,  f).  But  its  complete  expression  is  not 
required,  because  $0,  T0,  W0  are  its  partial  differential  coefficients  at  the 
point  P  (r,  0,  0).  It  is  therefore 

+ (6) 


191,  192]  Method  of  Gauss  217 

and  the  terms  which  do  not  contain  f  can  be  neglected.     Thus  F0  is   £2 
simply.     Let  the  transformation  scheme  of  §  186  be  written 

x  =  l£  +  mrf  +  n^,     #1  =  liT  +  a'e' 


,     Zj,  =  I3r 

with  the  usual  relations  of  an  orthogonal  substitution.     Then 
Fl  =  (xxl  +  yy,  +  zztf  -  (a  V  +  6  V) 
=  (a'e'x  +  rj-y-  (a*  a*  +  b'2y2) 
=  r2£2  +  2aYr  fa  -  b'*F0  +  b'*z* 

=  £  {£(r2  -  b'2  +  6%2  +  2tt/eVZ1)  +  2<n  (a'e'rm,  +  6%?w,)  4- 
with  neglect  of  terms  not  containing  ^.     Similarly 

F_,  =  #\z?  -  (zx,  -  xztfja'*z?  -  (zyl  -  yz$ 
The  last  term  does  not  contain  £  and  hence 
a'*i*l?F-i  =  a2  Ij  +  mr  +  n       -  a'e'z 


or 

F_,  =  [b'*l3% 

Thus  F^,  F0,  F^  are  now  expressed,  as  far  as  necessary,  in  terms  of  f,  rj,  %. 
It  remains  to  calculate  Hl  and  H.2,  and  then  the  simple  comparison  of  the 
coefficients  of  p,  gij,  g£  in  (5)  and  (6)  gives  S0,  TQ,  W0. 

It  must  be  understood  that  it  is  not  the  object  here  to  obtain  the  most 
practical  form  of  calculation  in  its  final  shape,  but  rather  to  explain  the 
mathematical  principles  involved  and  to  be  content  with  showing  how  the 
computation  might  be  carried  out.  The  method  was  not  developed  by 
Gauss  in  the  complete  form  which  is  necessary  for  practical  computations. 
This  was  done  by  Hill.  The  introduction  of  elliptic  functions  in  the  modern 
form  is  due  to  Halphen. 


CHAPTER  XVIII 

SPECIAL     PERTURBATIONS 

193.  In  Chapter  XV  some  explanation  has  been  given  of  the  various 
classes  into  which  planetary  perturbations  naturally  fall  when  regarded  from 
a  practical  point  of  view.  There  is,  however,  another  kind  of  distinction 
which  can  be  drawn  among  perturbations,  depending  on  the  mode  of  calculation 
and  expression.  When  they  are  expressed  in  an  analytical  form,  from  which 
their  values  can  be  deduced  for  any  time  simply  by  giving  t  its  appropriate 
value,  they  are  called  absolute  perturbations.  For  all  the  major  planets 
a  theory  has  been  developed  in  this  form.  But  such  a  theory,  if  it  is  to  be 
complete  and  accurate,  demands  immense  labour,  which  is  justified  if  positions 
of  a  planet  are  constantly  required.  Moreover  questions  of  general  theory 
must  nearly  always  be  based  on  analytical  forms.  On  the  other  hand  there 
are  bodies  which  are  observed  during  one  short  period  only,  like  the  majority 
of  comets,  or  at  relatively  long  intervals,  like  the  periodic  comets.  In  such 
cases,  which  include  also  the  orbits  of  the  minor  planets,  the  method  of 
quadratures  is  resorted  to,  partly  in  order  to  save  labour  and  partly  to  avoid 
difficulties  which  have  not  hitherto  been  surmounted  by  analysis.  Perturba- 
tions calculated  in  this  way  are  called  special  perturbations.  The  advantage 
of  the  method  is  that  it  is  generally  applicable,  though  against  this  must  be 
set  the  frequent  necessity  of  continuing  the  calculation  without  a  break 
through  long  intervals  when  no  observations  have  been  made,  and  the  im- 
possibility of  making  any  general  inference  as  to  the  motion  outside  the  actual 
period  covered  by  the  computations.  There  are  exceptions  to  this  statement, 
because  important  researches  have  been  made  with  success  into  the  origin  of 
comets  by  the  method  of  special  perturbations,  and  the  periodic  solutions  of 
the  problem  of  three  bodies  have  also  been  largely  investigated  by  the  method 
of  quadratures.  But  generally  the  services  of  this  method  have  been  of  a 
practical  rather  than  a  theoretical  kind. 

The  method  of  quadratures  involves  an  arithmetical  technique  with  which 
the  reader  may  not  be  familiar.  It  therefore  lies  strictly  outside  the  intended 
scope  of  this  work,  which  is  not  concerned  with  the  actual  details  of  practical 
calculation.  But  the  computation  of  special  perturbations  fills  so  large  a  place 
in  the  practice  of  astronomy  at  the  present  time  that  it  cannot  be  dismissed 


193,  194] 


Special  Perturbations 


219 


without  some  description.  Accordingly,  in  order  to  interrupt  the  treatment 
of  dynamical  questions  as  little  as  possible,  a  brief  account  of  the  algebra  of 
difference  tables  is  given  in  the  final  chapter  of  the  book,  and  the  results  will 
be  quoted  here  without  proof. 

194.  Let  yn  be  a  tabulated  function  of  the  argument  t=a  +  nw,  where  n 
represents  a  series  of  consecutive  integers  and  w  is  a  constant  tabular  interval. 
As  the  practical  formulae  of  quadrature  depend  on  central  differences,  it  will 
be  convenient  to  represent  the  difference  table  thus : 


yn 


Kyn 


Here  yn  is  tabulated  in  a  vertical  column  and  the  successive  differences  on 
the  right  are  formed  directly  in  the  usual  way.  Thus  Ayn  =  ynJrl  —  yn,  and 
the  commutative  operator  K,  which  is  clearly  appropriate  to  central  (or  hori- 
zontal) differences,  represents  a  move  two  places  to  the  right  on  a  horizontal 
line  of  the  table.  Similarly  K~l  represents  a  horizontal  move  two  places  to 
the  left.  Two  columns  are  shown  on  the  left  of  the  tabulated  function,  and 
these  are  known  as  the  first  and  second  summation  columns.  The  relation  of 
each  to  the  adjacent  columns  on  the  right  is  precisely  the  same  as  that 
holding  between  any  two  consecutive  difference  columns.  Thus  the  first 
summation  column  contains  the  differences  of  the  second,  and  the  differences 
of  the  first  are  the  successive  values  of  the  function  itself.  The  first  column 
can  therefore  be  based  on  an  arbitrary  constant  and  formed  in  the  downward 
direction  by  adding  the  numerical  values  of  the  function  successively.  The 
second  summation  column  is  based  on  a  second  arbitrary  constant  and  formed 
from  the  first  in  the  same  way. 

The  table  thus  constructed  has  alternate  blank  spaces.  These  are  now 
filled  by  the  insertion  of  the  arithmetic  means  of  the  entries  standing  im- 
mediately above  and  below  each  space.  In  its  completed  form  the  table  may 
be  represented  thus : 


Kyn 
[k'Kyn\ 


where  the  mean  differences  are  distinguished  by  k  to  the  right  of  a  simple 
difference  or  by  k'  below  a  simple  difference.     As  a  matter  of  fact, 


but  for  the  immediate  purpose  in  view  these  operators  serve  merely  to  define 
the  position  of  entries  in  the  difference  table.     They  are  all  algebraic. 


220  Special  Perturbations  [CH.  xvm 

195.  The  formulae  available  for  executing  the  necessary  quadratures  can 
now  be  given.  Numbered  as  in  the  last  chapter  of  the  book,  to  which 
reference  can  be  made  for  proofs,  they  are  these : 

1        "   "       191     -?+. ..V.    ...(28) 


a+mw  /  1  17 


a+nw 

-"  .........  (30) 


a+mwr  rx 

L/i 


where  m  is  written  in  the  upper  limit  in  the  place  of  n  +  £.  The  commutative 
operator  k  must  of  course  be  carefully  distinguished  from  the  Gaussian 
constant  k. 

The  lower  limits,  b  and  c,  are  arbitrary  and  correspond  with  the  arbitrary 
constants  involved  in  forming  the  first  and  second  summation  columns.  If 
the  lower  limit  is  to  be  c  —  a, 


which  fixes  one  constituent  of  the  first  column,  and  the  rest  follow.     If  the 
lower  limit  is  to  be  c  =  a  +  %w, 

17      ,        367  \ 

r-  +  ...»  .........  (27) 


Similarly,  if  the  lower  limit  b  of  the  second  integration  is  a, 


and  the  value  of  this  particular  constituent  makes  the  whole  of  the  second 
summation  column  determinate.     If  the  lower  limit  is  b  =  a  +  ^w, 

'  +  *'      -       K  +          K  *  -  '  •'  *•  '  -(33) 


In  general,  6  =  c  and  (29)  and  (32)  are  used  together,  or  (27)  and  (33).     In 
the  latter  case  (33)  may  also  be  written 


In  whatever  way  the  lower  limits  are  determined,  (28)  and  (30)  will  give  the 
integrals  to  the  upper  limit  a  +  nw,  and  (26)  and  (31)  to  the  upper  limit 


w. 


195-197] 


Special  Perturbations 


221 


196.  The  application  of  quadratures  to  the  solution  of  differential  equa- 
tions such  as  arise  in  dynamical  problems  can  be  explained  by  a  simple  but 
fairly  general  form.  Consider  the  equation 


or,  as  it  may  be  written, 

Hence,  by  (30), 

x  =  w2  (wD)~2X 


or 


(1) 


Now  suppose  that  we  have  a  solution  in  progress,  giving  at  a  certain  stage, 


tn 


Kxn 


Xr, 
X* 


KXn 


KX 


Here  Xn  is  a  known  function  of  xn  and  tn.  It  is  required  to  find  xn+3  and  Xn+s 
which  depend  on  tn+3  and  on  one  another,  so  that  they  cannot  be  calculated 
directly.  For  simplicity  the  time  interval  w  may  be  imagined  to  be  so  small 

that  -— -  K-Xn+l  is  negligible.     The  general  run  of  the  differences  KX  will 

suggest  a  close  guess  to  the  value  of  KXn+2,  though  the  true  value  requires 
a  knowledge  of  Xn+3  and  therefore  of  xn+3  itself.-  This  leads  to  a  correspond- 
ing provisional  value  of  Kxn+z  by  (1)  and  hence  to  xn+3  —  xn+^  or  xn+3.  Then 
Xn+3  can  be  calculated,  in  general,  with  the  accuracy  which  is  finally  necessary. 
If  this  be  so,  KXn+2  is  now  accurately  known,  and  hence  xn+z  by  a  simple 
repetition  of  the  same  process,  in  which  if  need  be  an  allowance  for  K*X  can 
be  made.  After  every  few  steps  in  the  calculation  the  whole  can  be  rigorously 
checked  by  the  difference  formula  (1)  and  either  verified  or  corrected  if 
necessary.  In  general  small  corrections  of  xn  do  not  entail  a  re-adjustment 

Of  Xn.    " 

197.  This  is  the  principle  of  the  method  employed  by  Cowell  and 
Crommelin  in  calculating  the  path  of  Halley's  Comet  during  the  two  revolu- 
tions 1759-1835-1910.  It  is  the  crudest  possible  method  in  the  sense  that 
it  ignores  completely  what  is  known  of  the  approximate  orbit  and  is  based  on 
the  equations  of  motion  in  their  primitive  form,  but  it  is  none  the  less  ex- 
tremely effective  for  its  practical  purpose.  The  origin  of  coordinates  is  taken 


222  Special  Perturbations  [CH.  xvm 

at  the  centre  of  gravity  of  the  solar  system,  with  the  axis  of  x  towards  the 
equinox,  the  axis  of  y  towards  longitude  90°  and  the  axis  of  z  towards  the 
N.  pole  of  the  ecliptic  for  a  stated  fixed  epoch.  The  equations  of  motion  are 
then  (§  20) 

dU  dU  dU 

mx  =  —  •=—  ,  my  =  —  ^—  ,  mz  =  --  r- 
dx  dy  dz 

where 

tf  =  -  #m  2  n,j  {(x  -  Xjf  +  (y-  ytf  +  (z-  Zj}z]  ~  * 

and  2  includes  the  Sun  and  all  the  disturbing  planets.  Thus  the  typical 
equation  may  be  written 


where 

X  =  -  2  (fcw*mj)  (x  -  Xj}  rf* 

and  kzw2mj  is  a  constant  for  each  attracting  body.  The  problem,  being  in 
three  dimensions,  involves  the  parallel  solution  of  the  three  similar  equations 
for  x,  y  and  z.  It  is  convenient  to  change  the  time  interval  from  time  to  time 
according  to  circumstances,  in  order  to  economise  labour  in  computing  the 
forces  by  making  the  interval  as  long  as  experience  may  show  to  be  practicable. 
In  the  example  referred  to,  w  =  2p  days,  where  p  has  integral  values  ranging 
from  1  in  the  neighbourhood  of  the  Sun  to  8  in  the  most  distant  part  of  the 
orbit.  As  the  comet  recedes  from  the  Sun  it  becomes  feasible  to  treat  first 
Venus  and  later  the  Earth  and  Mars  as  forming  a  centrobaric  system  with 
the  Sun,  so  that  the  separate  computation  of  their  attractions  is  avoided. 
The  solution  is  started  by  deriving  the  rectangular  coordinates  of  the  comet 
on  two  consecutive  dates  from  the  osculating  elements  at  the  intermediate 
epoch  1835. 

A  similar  treatment  has  .been  applied  to  the  path  of  Jupiter's  eighth 
satellite,  which  is  so  distant  from  its  primary  that  the  solar  perturbations  are 
relatively  very  considerable. 

198.  The  above  process  is  closely  related  to  the  more  usual  method  of 
calculating  special  perturbations  in  rectangular  coordinates,  which  dates  from 
Encke.  Here  the  origin  is  taken  at  the  centre  of  the  Sun  and  a  fixed  ecliptic 
system  of  axes  is  generally  chosen.  Let  {x,  y,  z)  be  the  position  of  the 
disturbed  body  P,  (xj,  yj,  Zj)  of  the  typical  disturbing  planet  Pj,  and  let 
SP  =  r,  SPj  =  PJ  and  PPj  —  Aj.  Then  the  equations  of  motion  of  P  relative 
to  the  Sun  are  of  the  form  (§  23) 

Qj    CC  Tn/-i  \      ™  To^-*  i^j    ~" 

—  =  -  k2  (1  +  m)  -  +  &  2  mt    ^ 
dt2  r3  }  \   A/ 

But  the  undisturbed  motion  is  given  by 


-*•<' 


197,  198]  Special  Perturbations  223 

where  (#„,  y0,  ZQ)  and  r0  can  be  calculated  at  regular  intervals  of  time  from  the 
osculating  elements.  Hence  if  (£,  rj,  £)  are  the  perturbations,  where 

g  =  X  XQ  ,    ... 

&%      72  fv       /«/  —  #      Xj\  ,  /a?0       # 

-r^  =  A;2  ^2m,-    -^r-  ---  *    +  *  +  m)    ^  ~  ^ 
efo2          I         V  A/       p//  Vn3     r 

The  right-hand  side  contains  (£,  77,  £)  implicitly,  and  therefore  extrapolation 
is  necessary  as  in  §  197.  But  in  the  first  member  £,  which  is  of  the  first 
order  in  m,-,  is  multiplied  by  nij  and  hence  if  the  second  order  in  rrij  be 
neglected  (a-0,  y0,  z0)  can  be  directly  substituted  for  (ac,  y,  z}.  This  is  conse- 
quently known  as  the  direct  member,  but  it  is  quite  possible  to  include 
approximate  values  of  the  perturbations  as  they  become  known  in  the  course 
of  the  work,  and  thus  to  make  allowance  for  the  higher  orders  of  the  disturb- 
ing masses.  The  second  member,  which  has  been  called  the  indirect  member, 
has  no  small  multiplier  and  besides  is  expressed  as  the  difference  of  two 
nearly  equal  quantities.  To  avoid  this  inconvenience  the  transformation 

rt*2  /y«   3  n 

5-1+2*  ^=(i+2<2r*=i-/? 

'0  ' 

is  made,  where 

0  +  4  17)  17  +  (*.  -I- 
5.7.9 


...      ...............  (2) 

and  /  is  tabulated  as  a  function  of  q,  which  is  a  small  quantity.     The  equation 
for     now  becomes 


h%  ..........................................  (3) 

with  parallel  equations  for  rj  and  £.  This  treatment  is  not  applied  to  the 
planets  with  sensible  masses,  but  only  to  bodies  whose  masses  are  negligible 
and  generally  unknown.  Hence  h  =  k2r0~3. 

Suppose  that  n  —  1  steps  in  the  quadrature  have  been  carried  out,  so  that 
£»»-!>  £w-i  ai>6  known  and  gn  is  required.  As  in  §  197  w2  can  be  omitted  by 
substituting  w*k*  for  fc2.  Then,  by  (SO), 


or 

1 
'•  Sx,n  + -•{*  hfqxn    (5) 

1Z 


224  Special  Perturbations  [OH.  xvm 


Here  SXjn  comprises  the  terms  which  can  be  directly  calculated,  for  2. 
represents  the  direct  terms,  K~*$n  follows  from  the  previous  stage  of  the 
quadrature,  and  K%n  can  be  extrapolated  easily  owing  to  its  small  multiplier. 
Also  xn=x0+l;n  is  known  well  enough  since  it  is  multiplied  by  q.  But  q  itself 
is  not  accurately  known.  By  combining  the  three  parallel  equations  of  the 
same  type  as  the  last  with  the  above  equation  for  q,  it  follows  that 

1     \         /         I      \  1         --/        1  •    \ 

where  S  refers  to  the  three  coordinates.     Thus,  f  being  easily  extrapolated, 
q  can  be  calculated.     The  combination  of  (3)  and  (5)  now  gives 

—  Sx,n) 

whence  £n  can  be  calculated,  and  therefore  £n  by  (4).     Thus  the  quadrature, 
once  started,  proceeds  step  by  step. 

In  order  to  start  the  quadrature  the  four  dates  are  taken  such  that  the 
epoch  of  osculation  coincides  with  the  centre  of  the  middle  interval.  With 
£  =  0  the  direct  terms  in  (•  are  calculated  and  the  difference  table  is  formed. 
By  applying  (27)  and  (34)  approximate  values  of  £  are  obtained  whereby  the 
indirect  terms  can  be  brought  in.  The  process  is  then  repeated  until  the 
final  approximation  is  reached.  The  rest  of  the  calculation,  giving  the  results 
by  means  of  (30),  has  already  been  explained. 

199.  Special  perturbations  may  also  be  directly  calculated  for  polar 
coordinates.  Let  the  cylindrical  coordinates  of  the  disturbed  mass  m  be 
(p,  6,  z),  the  fundamental  plane  being  the  plane  of  the  osculating  orbit  itself 
at  the  epoch  t0,  and  the  initial  line  passing  through  the  ecliptic  node.  The 
rectangular  coordinates  of  the  typical  disturbing  planet,  of  mass  rrij,  relative 
to  the  Sun  are 

Xj  =  TJ  cos  BJ  cos  Lj ,    yj  =  TJ  cos  Bj  sin  Lj ,     Zj  =  TJ  sin  Bj. 

The  kinetic  energy  of  m  is  \m  (p2  +  p262  +  z'2\  and  therefore  the  equations  of 
motion  are,  since  r2  =  p2  +  z2, 

d2o        /dO\2  dR 


/d6\dR       d*z  dR 


/\_  *z_ 

dt\p  dt)~d8'      dt*~ 
where  (§23) 


ij  {  Aj-1  —  rf3  [prj  cos  Bj  cos  (Lj  —  0)  +  zrj  sin 
A/  =  pz  +  z*  +  rf  -  2  [prj  cos  Bj  cos  (Lj  -0)  +  zrj  sin  Bj 


198-200]  Special  Perturbations  225 

Hence 

p  -pfr  =  -  k2  (1  +  w)  pr~3  -  k2  2m,  {p&f-*  -  (  A/-»  -  r/-3)  r}  cos  B5  cos  (L5  -  6)} 
d  (p2  6)1  dt  =  k2p  2m,-  (  Af8  -  rf3)  r,  cos  £,  sin  (L,  -  6) 

z  =  -k2(l+m)  zr~3  -  kz  2m,-  [z&f*  -  (A,-~8  -  rf)  r,-  sin  Bj}. 
Let 


where  /  is  the  same  function  of  q  as  in  (2)  and  can  usually  be  replaced  by  3 
simply,  because  z  is  merely  the  perturbation  in  latitude  reckoned  from  the 
osculating  plane.  The  equations  of  motion  can  now  be  written  : 

p-p62  +  k2  (1  4-  m)  p~2  =  pH 


where 

H  =  $&  (1  +  m)fp~5z2  +  k2  2m,  {p-1  (bf*  -  r/"3)  r,  cos  ^  cos  (Lt  -6)-  A,-"3} 
f/  =  k2p  2m,-  (Aj-3  -  r/-»)  r,-  cos  £_,-  sin  (J^  -  <9) 
F!  =  k2  2m,-  (Aj-3  -  r,-3)  r5  sin  5,-  +  P2  (1  +  m)fp~5z3 
W2  =  k2  2m*  A;--3  +  A;2  (1  +  m)  /j"3. 

The  third  equation  is  now  in  the  required  form  to  determine  z.     The  first 
two  must  be  transformed  in  order  to  obtain  p  and  6. 

200.     The  second  equation  gives 

Udt 

t0 

where  h  is  the  undisturbed  constant  of  areas,  so  that 
h  -  {k2  (1  +  ra)  p0}*  =  n0aQ2  cos  <f>0 

po>  n0,  a0,  sin<£0  being  the  osculating  parameter,  mean  motion,  mean  distance 
and  eccentricity.     Hence 


,«  r«,r        rt 

p-*dt  +       \p-2      Udt    dt 
J  «0  J  tt  L       Jt 


rt 

Ud 
J 


=  &>0  +  V  +  Aft) 

where  00  is  the  initial  value  of  6  and  «o0  is  the  distance  of  the  undisturbed 
perihelion  from  the  node.  The  angle  A&>,  which  represents  and  is  defined  by 
the  double  integral,  would  vanish  in  the  absence  of  disturbing  forces.  In  the 
same  circumstances  V  would  be  the  undisturbed  true  anomaly.  Thus  V  may 
be  regarded  as  the  disturbed  true  anomaly  and  Ao>  as  a  rotation  of  the  apse. 

In  the  rotating  orbit  thus  defined,  in  which  the  elements  p0,a0,e0,  <£0  keep 
their  osculating  values,  let  p  (1  +  v)~l  be  the  radius  vector  corresponding  to 
the  true  anomaly  F,  so  that,  since  V—  hp~2, 

1  +  e0  cos  V  =  p0  (1  +  v)  p~l 

-  e0  sin  V=h~lp2p0  {-  (1  +  v)  p~2p  +  vp~1} 

-  e0  cos  V-  h~-p2pQ  {-  (1  +  v)  p  +  pi/}. 

P.  D.  A.  15 


226  Special  Perturbations  [CH.  xvm 

Hence 

-  p)  +  h-2p0p3v 


or 

p  =  h?p~3  +  (1  4-  v)~*pv  -  tf  (1  +  m) 
But 


•   'o 

Therefore,  by  the  first  equation  of  motion  in  the  form  last  found, 

pH=(l  +  i/T1  PV  +  &2  (1  +  m)  (1  +  v)-lvp~*  -  p~3  f 

J  «« 

which  can  be  written  in  the  form 

v  +  H  2  v  =  H1 
where 


From  this  equation,  which  is  of  the  same  form  as  that  in  z,  v  can  be  found 
by  mechanical  integration. 

Again,  instead  of  finding  F  by  a  direct  quadrature,  the  necessary  correction 
N  is  found  to  the  mean  anomaly  calculated  with  the  undisturbed  mean  motion 
n0,  so  as  to  reproduce  the  true  anomaly  V  or  the  eccentric  anomaly  E  in  the 
rotating  orbit.  Thus 

E  -  e0  sin  E  =  M0  +  n0  (t  -t0)  +  N 
a0  (1  -  e0  cos  E)  =  p(l  +  i/)-1. 
Hence,  by  (7)  of  §27, 

ft  +  HO  =  (  1  -  e0  cos  E)  E  =  pa0~l  (1  +  v)~l  V  .  dE/d  V 
p          h    1  —  e0  cos  E  _      n0 

~  a0  (1  +  v)  '  p2  '      cos  <£„      ~  (T+  i/)3 
and 


201.     The  whole  problem  is  therefore  reduced  to  the  mechanical  solution 
of  the  equations 

d^v  dN_  2  +  v 

dt*+  ~dt=     'W°^TTi2 


dt 

When  v,  N,  A&>,  z  are  known,  the  coordinates  r,  6  and  the  latitude  X  are 
given  by 

E  —  e0  sin  E  =  Jf0  +  n0  (t  —  t0)  +  N 

p  sin  V  =  (1  +  v)  a0  cos  <f>0  sin  E,     p  cos  V=(l+  v)a0  (cos  E  —  e0) 


SOD-SOS]  Special  Perturbations  227 

Perturbations  to  the  first  order  will  be  obtained  by  calculating  the  quanti- 
ties occurring  in  the  differential  equations  according  to  the  osculating  elements, 
but  as  they  become  known  in  the  course  of  the  work  their  approximate  effect 
on  the  coordinates  of  the  disturbed  planet  can  be  introduced  before  integra- 
tion. The  integral  of  U,  and  also  N  and  A&>,  can  thus  be  found  by  direct 
quadrature  by  applying  (27)  and  (28).  For  v  and  z,  which  require  exactly 
similar  treatment,  the  case  is  slightly  different.  As  before,  the  time  interval 
w  is  removed  by  writing  w2k'2  for  k2,  which  is  equivalent  to  making  this  interval 
the  unit  of  time.  Then  at  any  stage  n,  when  zn^  and  K~lzn  are  known, 

y  —  W—W? 

&n  —  "  i        "  2^n 


w*  -  w,  (i  +  1  IT,)"  {(*•-  -  4  K  )  1,  +  1 

and  this  last  equation  will  determine  zn   with  the  needful  accuracy,  and 
hence  zn  and  K~*zn+l  for  the  next  stage. 

This  method  is  due  in  principle  to  Hansen.  The  perturbations  start  from 
zero  values  and  remain  small  for  a  considerable  length  of  time.  This  conduces 
to  accuracy  and  is  an  advantage.  The  method  is  less  simple  than  that  of 
rectangular  coordinates,  and  for  the  easier  construction  of  an  ephemeris 
requires  the  determination  of  new  osculating  elements  by  a  process  which  is 
itself  complicated  and  is  omitted  here.  Perturbations  of  the  coordinates  are 
recommended  by  the  fact  that  there  are  three  coordinates  as  against  six 
elements  to  be  determined  by  quadratures,  and  their  computation  is  suitable 
for  practical  needs  in  the  case  of  a  body,  such  as  a  periodic  comet,  which  can 
only  be  observed  at  relatively  long  intervals.  Otherwise  it  is  preferred  to 
perform  the  calculation  on  the  elements  directly. 

202.  With  slight  changes  which  will  be  readily  understood  the  equations 
found  in  §  142  for  the  perturbations  of  the  elements  may  be  written  : 

dijdt  =  rW  cos  u/k^/p 
d£l/dt  =  rW  sin  ujk\/p  sin  i 
d<f>/dt  =  [S  sin  v  +  T  (cos  v  +  cos  E)}  *Jpjk  cos  </> 
dvr/dt  =  {—  pS  cos  v  +  (p  +  r)  T  sin  v  +  r  W  sin  <f>  tan  A  i  sin  u]  /k\/p  sin  <j> 

dnjdt  =  —  3  (rS  sin  <£  sin  v  +  pT)  cos  <J>/pr 

/t  ftn 
-r.dt 
.!„  <M 

15—2 


228  Special  Perturbations  [CH.  xvm 

where  v  represents  the  true  anomaly  and  m  is  neglected,  so  that  /*  =  k*.     Let 

wS  =  kF,  Vp,     wT  =  kF2  Jp,     wW  =  kF3  *Jp. 
Then  the  equations  are  of  the  form 

wdijdt  =  [i,  3]F3,    w  dfl/dt  =  [H,  3]  F. 

wd<}>{dt  =  [</>,  1]  F1  +  [</>,  2]  F2,     wdnfdt  =  [n,  1]  Fl  +  [n,  2]  Ft 

wdvr/dt  =  [w,  1]  Ft  +  [or,  2]  F2  +  [or,  3]  ^3 


where 

[i,  3]  =  r  cos  w,     [12,  3]  =  r  sin  w/sin  t 

[<£,  l]=p  sin  v  sec  (/>,     [</>,  2]  =  ^  (cos  v  +  cos  .£')  sec  <f> 

[or,  1]  =  —  p  cos  v/sin  <£,     [or,  2]  =  (p  +  r)  sin  v/sin  0,     [or,  3]  =  r  sin  u  tan  £t' 
[Jf,  1]  =  -  {[or,  1]  +  2?-}  cos  </>,     [#,  2]  =  -  [w,  2]  cos  <j> 

[n,  1]  =  —  3k  sin  <£  cos  </>  sin  w/Vp,     [w,  2]  =  —  3k  cos  0  Vp/r- 

For  a  minor  planet  disturbed  by  Jupiter,  40  days  is  generally  found  a  suitable 
value  for  the  interval  w. 

The  disturbing  function  R  may  be  taken  in  the  form  found  in  §  199 
except  that  the  argument  of  latitude  is  now  u  =  v  +  or  —  H  instead  of  6. 
Thus 

R  =  k~  ^mj  (A,"1  —  rf3  [prj  cos  Bj  cos  (Lj  —  u)  +  zrj  sin  Bj~]} 

and  if  the  directions  of  the  components  S,  T,  W  be  recalled, 

„  _  dR        T  _  1  dR        u/.     dR 

®  —  ~^r~  5      •*•  —  ~  ~^~  >      "  —  "^  — 
op  p   du  oz 

where  after  differentiation  z  —  0,  because  the  plane  of  reference  is  the  plane 
of  the  instantaneous  orbit.     For  the  same  reason  p  =  r.     Hence 


Fl  =  p~    2  (kwmj)  {(Aj~3  -  rf3)  TJ  cos  Bj  cos  (Lj  —  u)  —  rA/~3} 
Fz  =  p  ~  *  2  (kwmj)  (  Aj~3  —  rj-3)  7*j  cos  5j  sin  (Lj  —  u) 
-  ?)-3)  ?)•  sin  Bj 


and 

Aj2  =  r2  +  ry2  —  2rrj  cos  Bj  cos  (Z>j  —  u). 

203.  Let  (,,  6j  be  the  heliocentric  longitude  and  latitude  of  the  disturbing 
planet,  which  with  log  r,-  are  given  in  annual  tables  like  the  Nautical  Almanac. 
The  relations  between  ecliptic  coordinates  (x,  y,  z)  and  the  orbital  coordinates 
(£>  ^  £)>  the  axis  of  f  passing  through  the  ecliptic  node,  are  shown  by 

x  y  z 

£  cos  fl  sin  n  0 

•  17         —  cos  i  sin  U        cos  i  cos  £1          sin  i 
£          sin  i  sin  U        —  sin  i  cos  n        cos  i 


202-204]  Special  Perturbations  229 

which  is  the  scheme  derived  in  §  65.     Hence 

£  =  cos  Bj  cos  LJ  =      cos  bj  cos  (^-  —  H) 

T)  =  cos  J3j  sin  LJ  =       cos  fy  cos  i  sin  (^  —  fl)  +  sin  bj  sin  i 

£  =  cos  Bj  =  —  cos  bj  sin  i  sin  (^-  —  O)  +  sin  bj  cos  t 

and  thus  F1}  F2,  Fs  can  be  calculated,  so  far  as  the  coordinates  of  any  disturb- 
ing planet  are  concerned. 

But  F1}  Fz,  Fs  and  the  coefficients  [i,  3],  ...,  involve  also  the  varying 
elements  and  coordinates  which  depend  on  them.  The  elements  may  be 
identified  with  the  osculating  elements  at  the  initial  epoch  £0  and  the  co- 
ordinates may  be  calculated  as  in  undisturbed  motion.  Then  the  result  of 
mechanical  integration  will  give  the  perturbations  of  the  first  order.  When 
these  are  known  for  the  several  dates  covered  by  the  work,  the  calculation 
can  be  repeated  with  the  improved  values  and  a  higher  approximation  can  be 
obtained.  The  work  can  be  arranged  so  as  to  obviate  this  repetition  by 
including  the  perturbations  to  date  at  each  step. 

204.  The  five  elements  i,  fl,  $,  OT,  n  require  only  a  single  quadrature. 
The  lower  limit  a  +  \w  is  made  to  coincide  with  the  epoch  of  osculation  and 
the  tables  are  formed  in  accordance  with  (27).  The  corresponding  perturba- 
tions are  then  given  by  (28)  or  (26)  according  as  a  +  nw  or  a  +  (n  +  ^)  w  is 
preferred  for  the  final  date.  It  is  to  be  noticed  that  the  differential  equations 
for  the  elements  have  been  reduced  to  a  form  in  which  w  occurs  explicitly  as 
a  coefficient  of  the  derivatives  on  the  left-hand  side.  It  will  disappear  when 
the  quadratures  are  effected,  its  function  being  to  make  the  unit  of  time  agree 
with  the  tabular  interval.  But  the  unit  of  time  is  not  really  changed,  and 
with  the  ordinary  Gaussian  constant  k  occurring  in  the  combination  kwnij  for 
each  disturbing  planet  remains  one  mean  solar  day.  Thus  the  perturbation 
in  n  which  will  be  drawn  by  this  process  will  be  the  increment  in  the  mean 
daily  motion.  Since  all  the  elements  are  in  the  form  of  angles,  it  is  con- 
venient to  express  k,  so  far  as  it  occurs  in  Fl}  Fz,  F3  through  the  combination 
kwnij,  by  its  value  in  arc  (log  k"  =  3'55...).  But  in  [w,  1],  [n,  2]  k  has  its 
purely  numerical  value  (log  k  =  8*235.  .  .). 

The  perturbation  in  M  can  be  conveniently  divided  into  two  parts.  The 
first, 

8,M  -  w~l  |  {[M,  1]  F,  +  [M,  2]  Fa}  dt 

is  calculated  in  precisely  the  same  way  as  the  other  five  elements.    The  second  is 

B2M= 


The  table  having  been  prepared  for  the  first  quadrature  on  the  basis  of  (27) 
and  (28),  the  second  can  be  performed  by  means  of  (34)  and  (30).     The 


230  Special  Perturbations  [CH.  xvin 

immediate  result  will  give  w~"82M,  which  must  therefore  be  multiplied  by  w2. 
To  avoid  this  large  multiplier  it  is  usual  to  calculate  w&n  from  wzdn/dt  at  the 
first  quadrature  (giving  the  increment  in  the  w-day  mean  motion).  This 
alters  the  time  unit  of  the  acceleration  and  therefore  no  multiplier  will  be 
required  by  82M,  a  result  which  can  be  otherwise  seen  by  noticing  that  all 
the  tabular  entries  are  multiplied  by  w,  while  the  integrand  is  divided  by  w, 
being  in  fact  dn/dt  instead  of  w.  dn/dt  as  in  the  first  quadrature  actually 
performed  on  this  plan. 

205.  In  the  case  of  parabolic  and  nearly  parabolic  orbits  some  modifica- 
tion is  necessary.  The  equations  for  i,  H  and  ts  remain  valid,  except  that  it 
is  well  to  replace  </>  by  e.  The  equation  for  e  itself  becomes 

wde/dt  =  [e,  1]  F1  +  [e,  2]  F2 


e  \r     a 


But  the  equations  for  n  and  M  become  inconvenient,  if  not  illusory.  One 
suitable  substitute  is  easily  obtained  by  forming  the  equation  for  q,  the 
perihelion  distance.  Since  q  =  a  (1  —  e), 

do  da  de         2aw  ...        .  dn  de 

w  --^  =  (^  i  —  e)  w  -j —  aw  -j-  =  — ; —  (1  —  ?  *  — —  fi.9n  — 
at  dt  dt          3?i 

where 

I  V  >    -^'    I    """"    ^\  \  -*•     ~~    ^  /    I    *^)     A    I    ~~"    'v    I  t.     A    I 


=  — r  sin  <f>  cos  ^>  (1  —  e)  sin  v  —  ap  sin  w 
wp5 

=  2a2e  (1  —  e)  sin  v  —  a2  (1  —  e2)  sin  0  =  —  a2  (1  -  e)2  sin  v 

=  —  <f  sin  v 
and 


2a&    i  a»  (p      r\ 

-  »5(1  —  e)cos<f> ?-{•«- ) 

nr  r  e   \r     a) 


^      ap*     pr 
—  v)  —          i  — 
er       e 


pr     ao2  fl         2  pr       p3   . 

—    £_ £_     I I    «          __     -t  (  I    J»  g)     2 

e         r|_e      1+eJ       e      r  .e 


~~^'4  sin^-y  (1  +  e 


204-206]  Special  Perturbations  231 

Thus  a  valid  form  for  the  perturbation  of  q  is  obtained.  If  F1}  F2  have 
been  calculated  with  the  angular  value  of  the  constant  k  the  results  for  Se  and 
Sq  will  require  to  be  multiplied  by  sin  I". 

Again,  an  equation  can  be  formed  for  the  variation  of  T,  the  time  of 
perihelion  passage.  Since 

n(t-T)  =  M=e-vr+  I  ndt 

dn        dT      d  ,      dM      [dn 

(t  -  T)  -j-  -  n-j-   =  -r  (e  -  or)  =  -=-  -  I  -j-  dt 
'  dt         dt      dt^  dt      j  dt 

it  follows  that 


l(t~  T}  {[n'  1]  Fl  +  [1I>  2]  Fz}  ~  n~l  {[M'  l]Fl  +  [M'  2] 
=  [T,l]Fl  +  [T,2']Fa 
where 


[T,  1]  =  w-1  (t  -  T)  [n,  1]  -  n~l  [M,  1] 

v\ 
/ 


-e2)sin  v(t  -  T)     (1  -  e      (    ^      p  cos 
-f- 

^  n 


2  (1  -  erf  (        p  3ke   .       ,4      ^J 

v  —  ^  cos  v  --  1  sm  v(t-  T)> 

n          (       2e  2p*  J 

and 

[T,  2]  =  ?i-'  (t  -  T)  [n,  2]  -  n~l  [M,  2] 

(1  -  erf  (p  +  r)  sin  v 


nr  ne 

(I  -erf  (I,  8p* 


But  these  coefficients  are  in  a  form  absolutely  unsuitable  for  calculation, 
especially  in  the  case  of  a  parabola,  for  which  in  fact  they  are  required. 
The  difficulty  can  be,  and  is  best,  met  for  such  orbits  by  calculating  special 
perturbations  in  rectangular  or  polar  coordinates,  instead  of  directly  in  the 
elements. 

206.  The  reduction  of  [T,  1],  [T,  2]  to  a  calculable  form  is  not  altogether 
easy.  It  can  be  effected  in  the  following  way.  The  required  expressions  can 
be  written,  since  n2a3  =  k2,  p  =  a  (1  —  e2), 

2p*r      f-      cos  v  (1  +  e  cos  v)     3k(t-T)  } 

[r,l]— jryf- — iHl— -  S—  -esmn 

k  (1  -  e2)  1  2e  Z^r  ) 


Special  Perturbations  [CH.  xvm 

Now 

3 

k(t-T)=a?(E-esmE)  =  -£-  -  {(E  -  sin  E)  +  (I  -  e)  sm  E\ 

(1  -  e2)* 

»*       E-sinE  2»f 

=  '  tan  *v  +          cos2  *E  tan  iw 


(1  +  «)}  tan 
where 


- 

•"itan-i^cotf^"    - 
But  (§  27) 

r  cos2  i  w  =  a  (1  -  e)  cos*$E=p(I  +  e)~]  cos2 
and  therefore 


Let  [T7,  1],  [T,  2]  be  written  in  the  form 

r^  n_ 
" 


3 

rm  o-i         2pTr      (sin  v  (2  +  e  cos  v)     ,,} 
L-t,/J  =  ,    , i\  i~~         ' — n —  "~"  •*  «r 

where 

Fi  =  esinv.F,     F2  =  (l  -f  ecos?;)  F 
and  therefore 

Fj  cos  ^v  -  F2  sin  $v  =  —  (1  —  e)  sin  %v .  Y 

Hence 

F,  cos  $v  -  F2  sin  $v  =  -  $  sin  $v  sin  v  -    -  j(r— ^  .  2  tan2  ^t;)  -f  3} 


F,  sin  *  „  +  F2  cos  ^  =  -  i  cos  \v  sin  v    -.  .      ^? 

VI  +e    tan2^J£'/l 

+  |  cos  ^w  sin  v  {'2  (1  +  e)-1  tan2  \v  +  3}. 

The  expressions  involving  8  are  finite  and  they  are  multiplied  by  l-e,  which 
is  a  necessary  factor.     For  the  other  terms,  let 

y,  cos  ±v  -  yz  sin  ±v  =  -  1  sin  A  y  sin  v  .  ^^ 

I  +e 


yl  sn    v  +  y.>  cos  $v  =  f  cos  ^v  sin  v  +  (1  +  e)-1  cos  |v  sin  v  tan2  £w. 


206]  Special  Perturbations  233 

Then 

yi  =  %  (1  +  e)-1  sin2  v  (3e  +  tan2  $v) 

=  %  (1  +  e)-1  (1  -  cos  v)  [3e  (1  +  cos  v)  +  (1  -  cos  v)} 
<y2  =  (1  +  e)-1  sin  v  (f  (1  +  e)  cos2  £v  +  f  (1  —  e)  sin2  ^v  +  sin2  \v} 

=  -|(1  +  e)"1  sin  v  (4  —  cos  v  +  Be  cos  v). 
It  is  now  possible  to  write,  with  a  little  simple  reduction,  _  _ 

1    1  —  e  (  cos  v 

cos  2»  + 


2  •  T+-. 


—  (1  +  e  cos  0)  —  -  - 

and  yly  yz  have  been  determined  in  such  a  way  that 

1    1  —  e 

(Fj  —  y^cos^v  —  (F2  —  y2)  sin  ^v  =  — -.:r-    —  sinv.^rsin  6r 

(  F!  —  y^  sin  $v  +  (  F2  —  7/2)  cos  ^t>  =  —  ^ .  r-    —  sin  v .  g  cos  G 

where 

g  sin  G      !  —  $«.,  g  cos  G        S      2  tan4  At; 

-.2tan2iv,    ^-  -.--  -r^ 

sin  £v       1  +  e  cos  ^v       1  +  e 

Hence 

a 

cosw 


which  are  fairly  simple  forms,  but  still  require  the  calculation  of  g  sin  G,  g  cos  6r. 
In  the  limiting  case  of  the  parabola,  S  =  ^EZ  and 

g  sin  G  =  tan2  ^v  sin  ^v,    g  cos  (r  =  £  tan4  |v  cos  |t; 
which  then  completes  the  solution. 

The  more  general  case  of  a  very  eccentric  ellipse  can  be  related  to  the 
method  of  §  34.     In  the  notation  of  that  section, 

A 


A=—ri  -  :  —  FT,   T  =  tan  ,- 

9E  +  smE  I  -  £  A  +  C 

Hence 


15^     n  1-±A  +  C 

s=^—~    -**—  ^-j^  +  cyi  c\ 

1-%A'    tan2A^         l-§^      \5     A) 


234  Special  Perturbations  [CH.  xvm 

Now  by  the  method  of  §  34  A  (of  the  order  E'2)  is  found  in  calculating  v, 
and  C  (of  the  order  E*)  is  tabulated  with  argument  A.  With  the  same 
argument  it  is  possible  to  tabulate*  log  £  and  log  77,  where 


Then 

2  tan2  Aw  sin  A  v  ~     2  tan4  A 

- 


and  the  problem  is  thus  solved  in  a  practical  way.  Similar  treatment  can  be 
applied  to  hyperbolic  orbits. 

207.  It  sometimes  happens  that  a  comet  approaches  a  planet  (generally 
Jupiter)  so  closely  that  the  disturbing  force  due  to  the  planet  is  actually 
greater  than  the  force  due  to  the  solar  attraction.  It  is  then  more  convenient 
to  refer  the  motion  to  the  centre  of  the  planet  and  to  treat  the  solar  action  as 
the  disturbing  force. 

In  the  ordinary  case  the  equations  of  motion  of  the  comet  are  of  the  form 

d*x  ,  ,  ,  .  as  ln  (of  —  x  x'\ 
-rr=-  k2M  -  +  k2m  -—  1  ---  I 
dt2  r3  V  A3  ps/ 

where  M  is  the  mass  of  the  Sun,  m  the  mass  of  the  planet,  and  the  origin  is 
at  the  centre  of  the  Sun.  If  S,  P,  C  are  the  positions  of  Sun,  planet  and 
comet,  CS  =  r,  CP  =  A,  SP  =  p.  The  equations  involve  no  assumption  as  to  the 
relative  masses  of  the  Sun  and  planet,  and  if  they  are  interchanged  the 
equations  of  motion  of  the  comet  take  the  form 


where  the  origin  is  at  the  centre  of  the  planet,  so  that  #=#'  +  £,...,#'  +  £'=(),  — 
The  advantage  of  either  form  depends  on  the  ratio  of  the  total  disturbing 
force  to  the  corresponding  central  attraction,  and  it  will  rest  with  the  latter  if 

* 


that  is,  if  fjL  =  m/M,  when 


Let  GPS  =  0.    Then 

r  cos  CSP  =  p  -  A  cos  6 

r2  =  p2  -  2|oA  cos  0  +  A2. 

Now  in  the  nature  of  the  case  A  is  small  compared  with  p.     Hence 
r~4  =  p-*  +  4/>~5A  cos  0  +  2/>-6A2  (-1  +  6  cos2  0)+  ... 
r~3  =  p~s  +  3/o~4A  cos  0  +  f  /3~5A2  (—1  +  5  cos2  0)  +  . . . 

*  Bauschinger's  Tafeln,  Nos.  xxvn,  xxvin. 


206,  20?]  Special  Perturbations  235 

and  therefore 

r-4  +  p-4 _  2r-3/3-2 (p  -  A  cos  0)  =  p~6 A2 (1  +  3  cos2  6)+  .... 

To  gain  an  idea  of  the  planet's  sphere  of  influence  the  approximation  need  not 
go  further.  On  the  other  side  of  the  inequality  the  first  term  preponderates 
and  it  can  be  further  simplified  by  taking  r  —  p.  Thus  the  significant  terms 
of  the  lowest  order  in  A  give  the  inequality 

p-6A6(l  +  3  cos2  0)  <  /*yA-4 

and  the  polar  equation,  with  coordinates  (A,  6)  and  origin  at  the  centre  of 
the  planet, 

A  (1  +  3  cos2  0)™  =  p*p 

represents  a  meridian  of  the  bounding  surface,  which  is  one  of  revolution 
and  differs  little  from  a  sphere.  Its  radius  for  Jupiter,  Saturn  and  Uranus  is 
about  a  third,  and  for  Neptune  rather  more  than  half,  of  an  astronomical 
unit. 

When  the  comet  enters  this  sphere  of  influence  its  relative  coordinates 
(^  -  ar/,  yl  -  yj,  zl  -  z^}  or  (&,  tj1}  £)  and  its  relative  velocity  (&,  7)1,  £)  are 
known  and  its  orbit  about  the  planet  can  be  found,  with  the  constant  of 
attraction  &2m.  It  remains  within  the  sphere  so  short  a  time  that  the  solar 
perturbation  can  generally  be  neglected,  and  on  emergence  a  return  is  made 
to  the  heliocentric  orbit,  based  on  the  new  position  (|f2  +  #./,  772  +  y%,  £>  +  ^a')  or 
(a?2,  7/2,  22)  and  the  velocity  (x2,  y2,  4)- 


CHAPTER   XIX 

THE   RESTRICTED    PROBLEM   OF   THREE    BODIES 

208.  The  general  problem  of  three  bodies  is  reduced  to  a  relatively 
simple  and  ideal  form  when  two  of  the  masses  describe  circles  in  one  plane 
about  their  common  centre  of  gravity  and  the  third  body  has  a  mass  so  small 
as  not  to  affect  this  circular  motion  in  any  appreciable  degree.  Let  OXYZ 
be  a  set  of  rectangular  axes  rotating  with  angular  velocity  n  about  OZ,  OX 
following  OF,  and  let  the  coordinates  of  the  masses  //,,  v  be  (—  cl,  0,  0),  (c2,  0,  0) 
where  pc^  =  i/c2.  The  velocity  components  in  space  of  a  small  body  at  P  (%,  77,  £) 
are  (%  —  nv),  rj  +  n%,  £)  and  hence  the  kinetic  energy  of  unit  mass  is 


The  equations  of  relative  motion  are  therefore 


where  in  this  case 


dV 

— 
dr) 

8F 
a? 


pl}  p2  being  the  distances  of  P  from  p,  v.    The  result  of  adding  these  equations, 
multiplied  respectively  by  £,  17,  £,  gives  Jacobi's  integral  of  energy 

v2  =  p  +  #  +  £2  =  2  V  +  n2  (£2  +  vf)  -  C 
and  in  accordance  with  Kepler's  law 

&2  (p  +  v)  =  n*  (d  +  c2)3. 

209.     This  integral  has  a  very  simple  and  important  practical  application. 
Let  us  return  to  fixed  axes  through  //,,  so  that 

£  +  Cj  =  x  cos  I  +  y  sin  I,     r)  =  y  cos  I  —  a;  sin  I,     £=  z 
where  I  is  the  longitude  of  v  and  /  =  n.     Then 
£2  +  if  =  (x  +  ny)*  +  (y-  nx)* 
%-  +  rf  =  x*  +  yn-  —  2ca  (x  cos  1  4-  y  sin  1}  +  d2. 


208-210]       The  Restricted  Problem  of  Three  Bodies  237 

Hence  Jacobi's  integral  becomes 

£2  4.  if.  _j_  ^2  +  2n  (yx  —  ay)  =  2  V  -  2w2Cj  (x  cosl  +  y  sin  1)  +  ri*c?  —  C. 

The  special  circumstances  under  which  this  integral  can  be  usefully 
employed  are  these.  A  periodic  comet  between  two  appearances  in  the 
neighbourhood  of  the  Sun  may  pass  in  close  proximity  to  a  large  planet, 
Jupiter  for  example.  In  that  event  the  elements  may  be  so  altered  that  at 
the  second  return  the  identity  of  the  comet  is  doubtful.  At  times  when  the 
perturbations  are  small  and  the  heliocentric  motion  is  sensibly  elliptic, 


xy  —  yx  =     *p>p  cos 

the  latter  being  the  projection  of  the  areal  velocity  on  the  plane  of  the 
disturbing  planet.  Hence 

-  k'2/j,/a  —  2kn  V(/^>)  cos  i  =  2k2v/p2  —  2wad  (x  cosl  +  y  sin  1)  +  n?c?  —  C. 
It  is  supposed  that  the  change  in  the  observed  osculating  elements  takes 
place  almost  impulsively  within  the  region  of  the  planet's  influence.  This 
region  is  small  and  nearly  spherical.  Hence  /o2  is  the  same  at  the  beginning 
and  end  of  the  encounter,  and  the  changes  in  x,  y  and  I  are  small.  These  can 
be  neglected  together  with  the  other  planetary  perturbations,  and  therefore 
approximately 

/*/a'  +  '2k~ln  »J(np')  cos  i'  =  pfa"  +  2k-1  n  \'(pp")  cos  i" 

where  a,  a"  are  the  mean  distances  of  the  comet,  p'y  p"  the  parameters,  and 
i',  i"  the  inclinations  of  the  orbit  to  the  orbit  of  the  disturbing  planet,  before 
and  after  the  encounter.  For  the  Sun  //.=  1  and  A;2  (I  +i/)  =  w2a8,  where  a  is 
the  mean  distance  of  the  planet,  and  if  v  be  neglected 

a'-1  +  2a  ~  *  />'*  cos  i'  =  a"~l  +  2a  ~  *  p"**  cos  i" 

which  is  the  criterion  of  identity  proposed  by  Tisserand.  It  has  been  assumed 
that  the  orbit  of  the  disturbing  planet  is  circular,  but  some  allowance  can  be 
made  for  the  eccentricity  of  the  orbit  by  taking  into  account  the  actual 
motion  of  the  planet  at  the  time  of  the  suspected  encounter. 

210.     Let  the  problem  of  §  208  be  now  reduced  to  two  dimensions  (£=0). 
Then 

HP?  4-  vpf  =  H  (£  +  c,)2  +  /in*  +  v  (|  -  c2)2  +  vrf 

=  (H  +  v)  (f  2  +  r)*)  +  i*c*  +  vc.2*. 

Let  the  units  be  so  chosen  that  k=l  and  d  +  c2=  1,  with  the  consequence 
that  fju  +  v  =  n\  The  equations  of  relative  motion  may  now  be  written 

811 


238  The  Restricted  Problem  of  Three  Bodies      [CH.  xix 

where 

2H  =  p.  (Zpr1  +  />!«)  +  v  (2p*-1  -f  p2a) 

and  the  integral  of  relative  energy  is 

v2  =  2ft  -  C. 

These  are  the  equations  used  by  Sir  G.  H.  Darwin,  with  the  masses  /*  =  10, 
v  =  1,  in  his  researches  on  periodic  orbits.  Now  it  is  obvious  that  vz  cannot 
become  negative  under  any  circumstances.  Hence  the  curves  of  the  family 
given  in  bipolar  coordinates  by  the  equation 

2n  =  (7 

are  of  great  importance  in  the  restricted  problem  of  three  bodies,  because  they 
represent  barrier  curves  which  cannot  be  crossed  by  trajectories  characterized 
by  corresponding  values  of  C.  Thus  if  the  barrier  curve,  or  curve  of  zero 
velocity,  is  a  simple  loop  within  which  a  part  of  the  trajectory  lies,  then  the 
trajectory  can  never  pass  outside.  If  the  lunar  theory  can  be  compared  with 
this  simpler  problem  it  is  found  that  the  orbit  of  the  Moon  lies  within  such  a 
closed  curve  surrounding  the  Earth,  and  therefore  the  Moon  cannot  recede 
beyond  a  certain  limiting  distance  from  the  Earth.  This  remark  is  due 
to  Hill. 

The  simplest  view  of  the  general  character  of  the  curves  of  zero  velocity 
is  gained  by  considering  them  as  the  contour  lines  of  the  surface 


If  the  axis  of  z  is  taken  vertically  upwards,  and  motion  for  a  given  value  of  C 
is  supposed  to  take  place  on  the  actual  contour  plane  z  =  C,  then  it  is 
evidently  restricted  to  those  parts  of  the  plane  which  lie  underneath  the 
surface,  since  elsewhere  in  the  plane  the  velocity  becomes  imaginary.  Now 
the  main  features  of  the  surface  are  easily  represented  topographically.  At 
the  points  where  the  masses  /z,  v  are  situated  the  surface  rises  to  infinity,  but 
in  the  neighbourhood  of  these  singular  points  may  be  treated  as  two  peaks. 
At  any  considerable  distance  from  them  the  terms  pp?  +  vpf  are  predominant, 
and  the  surface  rises  indefinitely  in  all  directions.  Now  2ft  may  be  expressed 
in  the  form 

20  =  3  O  +  v)  +  p  (Pl  -  I)2  (1  +  Zpr1)  +  v  (p2  -  I)2  (1  +  2/ir1) 

and  clearly  has  an  absolute  minimum  value  3  (//,  +  v)  when  pl  =  p2  =  1,  i.e.  at 
the  vertices  of  the  equilateral  triangle  on  the  line  joining  the  masses  //,,  v. 
These  points  represent  the  bottom  of  two  valleys,  and  a  simple  consideration 
of  the  continuity  of  the  surface  shows  that  these  valleys  must  be  connected 
by  three  passes,  one  between  the  two  masses  and  the  others  on  the  same  line 
but  on  opposite  sides  of  the  two  masses  and  separating  them  from  the  rising 
surface  as  it  recedes  in  the  distance.  If  it  be  added  that  the  highest  pass  is 


210,  2ii]       The  Restricted  Problem  of  Three  Bodies 


239 


that  which  lies  between  the  masses  and  the  lowest  is  on  the  other  side  of  the 
greater  mass,  the  general  order  of  development  of  the  contour  lines  should  be 
sufficiently  evident.  The  critical  curves  for  Darwin's  special  case,  /A  =  10, 
v=  1,  are  illustrated  in  fig.  7.  The  whole  is  symmetrical  about  the  line  SJ. 


Fig.   7. 

211.  The  points  at  which  the  ovals  coalesce  or  disappear  evidently 
correspond  to  critical  values  of  fl.  Take  v  <  /*.  The  critical  values  are 
given  by 

9fi     911  opi     911  9/32     ... 
9|r     9/Oj   9^     9p2   9g 

911     9Ii  9/Oi     9O  9^2     /\ 
dtj      dpi   drj      9p2   877 

which  show  immediately  that  such  points  are  points  of  relative  equilibrium 
for  the  third  body.     These  equations  are  satisfied  in  the  first  place  by 


or  p1  =  p2  =  1.    This  gives  the  "  equilateral "  points  mentioned  above,  where 
is  an  absolute  minimum.     But  other  solutions  are  given  by 


=  0 


or  T?  =  0,  together  with  3fl/3|  =  0.    This  will  lead  to  the  three  points  collinear 
with  the  masses.     For  the  first,  lying  between  the  masses, 


so  that 


^ 


240  The  Restricted  Problem  of  Three  Bodies      [CH.  xix 

This  is  a  quintic  in  p.,,  with  only  one  real  root.  The  actual  solution  in 
a  particular  case  is  easily  found  by  trial  and  error  from  the  first  expression. 
The  second  expression,  when  expanded,  gives 


and  to  the  same  order 

C  =  p(3+  3p»»  +  2/>23)  +  v  (20T1  +  P22) 
=  fju  (3  +  3a2)  +  i/cr1  (2  +  fa) 
=  fjt  (3  +  9a2  +  2a3). 
For  the  second  collinear  point,  on  the  further  side  of  the  smaller  mass  i/, 

_   1      ,  9Pl  _  8/>2  _     ,     1 

PI-    +P*>   af"af- 

and  hence 


/A         p^-p*       (l-pj)(l+pz)* 
again  a  quintic  in  pz  with  only  one  real  root.     For  the  approximate  solution 


and  to  the  same  order 

(7  =  ^(3  +  3paa  -  2p2*)  +  v  (2/Dr1  +  p22) 
=  /n  (3  +  So2)  +  i/a-1  (2  -  I  a) 
=  i»,  (3  +  9a2  -  2a3). 

For  the  third  collinear  point,  on  the  further  side  of  the  larger  mass  /*,, 

dpl     dp2 
A-l+ft,     g|  =  ^  = 

and  therefore 

v  _  _  pT^^Pi  _         (2  +  o-)2  (3<r  +  3<r2  +  o-3) 
ft  ~      /32~a  -  Pa  ~      (1  +  o-)2  (7  +  12o-  -1-  (5a2  +  tr3) 

where  pt  =  1  +  er,  p2  =  2  -I-  tr.     Hence 

v  _  -g-(12  +  24q-+19o-2+...) 
fi~       7 
and 


an,  212]       The  Restricted  Problem  of  Three  Bodies  241 

» 

which  shows  that 

-  7i/  -  7o3 


is  a  very  close  approximation.     The  approximate  value  of  C  at  this  point  is 
C=/*(3  +  3<j*)  +  i>(5  +  £<r) 
=  A*  (3  +  -W-a")  +  3/*as  (5  -  ^a8) 


When  v/fj,  =  3a3  is  small,  as  in  the  case  of  the  planets  compared  with  the 
Sun,  the  above  approximations  are  generally  more  than  sufficient.  In  the 
limiting  case  //,  =  v  and  the  arrangement  of  the  points  of  relative  equilibrium 
is  obviously  symmetrical  with  respect  to  the  rotating  masses. 

212.     Let  £=£0  +  ^,  f}  =  Va  +  y,  where  (£0,  T/O)  is  a  fixed  point.     The 
equations  of  motion  may  then  be  written 

x  —  2ny  =  fi10  +  fl^  -f  £lny  +  ... 

y  +  Znx  =  ft01  +  fluos  +  no22/  +  .  .  . 
where 


provided  H  is  regular  at  the  point  (£„,  rj0)  and  x,  y  are  not  too  large.  If 
(£o>  fjo)  is  a  point  of  relative  equilibrium,  or  as  it  has  been  called  a  point 
of  libration,  and  as,  y  are  very  small,  the  linear  equations 


x  — 

y  +  2nd;  =  £lux  +  £lQ2y 

are  obtained,  and  these  determine  the  nature  of  the  equilibrium  at  (£„•  Vo)- 
For  they  are  satisfied  by  the  solution 

x  =  h  cos  (mt  —  a),     y  =  k  cos  (mt  —  /3) 
provided 

-  2mnk  sin  $  =  (m2  +  O^)  h  cos  a  +  Mln  cos  /3 

2wnA;  cos  /3  =  (m-  +  n20)  /?,  sin  a  +  kflu  sin  y8 

2wm/i  sin  a  =  h£lu  cos  a  +  (m2  +  no2)  ^  cos  /S 

—  2mnh  cos  a  =  h£lu  sin  a  +  (m2  +  n^)  k  sin  y8. 

These  equations,  which  result  from  equating  coefficients  of  cos  mt,  sin  mt,  are 

equivalent  to 

(m2  +  Hgo)  A  sin  («  —  #)=      2rarc& 

Mlu  sin  (a  —  /3)  =  -  2mn^  cos  (a  —  /8) 
(?n2  +  no2)  k  sin  (a  —  /3)  =      2mwA. 

AHn  sin  (a  -  /3)  =  —  2wmA  cos  («  —  /3). 

There  are  only  three  independent  equations  here,  and  this  should  be  so 
because   the    only  quantities  which    can   be    determined   are    the   ratio   of 

P.  D.  A.  16 


242  The  Restricted  Problem  of  Three  Bodies      [CH.-XIX 

amplitudes  hjk,  the  difference  of  phases  a  —  /3,  and  in.  The  three  equations 
may  be  written 

A2  (m2  +  Ho,,)  =     k*  (m2  +  no2) 
nn  tan  (a.  —  /3)  =  —  2mw 
(m2  +  fla,)  (m2  +  nw)  =      4ma»a  +  fl2n 

and  these  determine  a  series  of  infinitesimal  elliptic  orbits  about  a  point  of 
libration  when  m  has  a  real  value.  With  certain  simple  developments  such 
a  series  can  be  traced  into  a  family  of  finite  periodic  orbits. 

213.     The  third  equation,  that  is  the  quadratic  in  m2, 

w4  -  m2  (4na  -  fla,  -  HOB)  +  ^20  ^02  -  flan  =  0 

decides  the  question  of  stability  and  may  be  examined  more  closely.  If  the 
roots  in  m2  are  complex  or  negative,  real  exponential  functions  of  the  time 
enter  into  the  disturbed  motion  and  equilibrium  is  unstable.  If  the  roots 
are  real,  but  of  opposite  sign,  an  unstable  mode  of  motion  is  associated  with 
a  possible  elliptic  mode  and  equilibrium  is  again  unstable.  Here  the  point 
is  surrounded  by  an  unstable  family  of  orbits  initially  elliptic.  This  is 
illustrated  by  the  collinear  points  of  libration.  For  it  is  easily  found  that 
when  77  =  0 

nn  =  o,    n^  =  ^(2pr8+  1)  +  v  (2/or*  +  1) 

so  that  ftao  is  positive.     Now  at  the  point  of  libration  between  the  masses 

9pi     9p2  _  -,      an  _  an 
P\  +  PZ  —  ^>     "at  +  x£  —  •»•>     ~     —  2 

°f  °$  °Pl          °P2 

and  therefore,  since  rj  =  0, 

i  an    i  an 


/i    i\     /      i\ 

=    —  +  —  I  .  M  f  i  --  :  1 

2    v/oj    />2/  pi2/ 


which  is  negative  since  pi  <  1.  Similarly  no2  is  negative  at  the  other  collinear 
points  of  libration.  Hence  at  these  three  points  the  absolute  term  of  the 
quadratic  in  m2  is  negative  and  the  roots  are  real  and  of  opposite  sign.  Each 
of  the  points  is  therefore  surrounded  by  a  family  of  unstable  periodic  orbits. 
It  has  been  suggested  by  Gylden  and  by  Moulton  that  the  phenomenon 
known  as  the  Gegenschein  is  due  to  sunlight  reflected  by  meteors  which,  in 
spite  of  the  instability,  are  temporarily  retained  in  the  neighbourhood  of  that 
centre  of  libration  in  the  Sun-Earth  system  which  is  opposite  to  the  Sun  and 
at  a  distance  of  about  938,000  miles  from  the  Earth. 

When  both  values  of  m2  are  positive  the  disturbed  motion  is  the  resultant 
of  two  elliptic  motions,  and  equilibrium  is  stable.  This  may  be  illustrated 
by  the  "  equilateral  "  centres  of  libration.  At  one  of  these 

an  =  an  =  _52n_  = 

dp!      dpz     dprfpz 

api  =  _ap2=i     jjgi    d<o2      V3 

a£~       3£       2'       dy        drj        :    2 


212-2H]       The  Restricted  Problem  of  Three  Bodies  243 

and  therefore 


_          , 

9'  3?       •'  ~        4    ^" 


Hence  the  quadratic  in  m2  becomes,  since  n2  =  /n  +  v, 

m4  -m2(p  +  v)  +  sg-ftv  =  0 
and  the  roots  are  real  and  positive  if 


an  inequality  which  is  satisfied  if  p.]v  is  25  or  greater.  In  that  case  the 
equilateral  centres  of  libration  are  surrounded  by  two  distinct  families  of 
stable  periodic  orbits  which  are  ellipses  in  their  elementary  form,  with  periods 
tending  to  STT/TW.  If  the  masses  are  more  nearly  equal,  the  roots  of  the 
equation  in  m2  are  complex,  and  no  such  periodic  orbits  exist. 

Since  the  masses  in  the  system  Sun-Jupiter  satisfy  the  condition  of 
stability,  and  the  disturbing  influence  of  Jupiter  predominates  over  the 
minor  planets,  it  might  be  expected  that  planets  would  be  found  in  this 
group  approximating  to  the  equilateral  configuration.  Such  planets,  with 
a  mean  motion  nearly  equal  to  that  of  Jupiter,  have  actually  been  discovered. 

214.  A  valuable  insight  into  the  general  character  of  the  solutions  of  the 
problem  of  three  bodies  is  obtained  from  the  periodic  solutions  because  they 
repeat  themselves  after  every  period.  These  solutions  have  therefore  been 
the  subject  of  much  laborious  study.  But  such  orbits  will  not  be  indefinitely 
permanent  unless  they  are  also  stable.  Hence  it  is  necessary  to  study  them 
in  relation  to  those  orbits  which  initially  differ  but  little  from  them. 

The  original  equations  of  motion  give 


or 

^  +  2nv2=-vd-^  =  vN  ...........................  (1) 

where  R  is  the  radius  of  curvature  of  the  orbit,  8p  is  an  element  of  the 
outward  drawn  normal,  and  N  may  be  called  the  component  of  effective 
force  along  the  inward  normal.  Hence  if  the  tangent  to  the  orbit  makes  the 
angle  <£  with  the  axis  of  £,  R  =  v/</>  and 


16—2 


244  The  Restricted  Problem  of  Three  Bodies      [OH.  xix 

Also  the  equation  of  relative  energy  gives,  when  the  constant  C  remains 

unaltered, 

dv      .      8H        vdv  _  an 
ds  ds  '       dp       dp  ' 

Let  the  undisturbed  orbit  at  P  be  defined  by  the  quantities  s  and  <f>,  and  the 
corresponding  point  P'  on  the  neighbouring  orbit  by  bs  along  the  undisturbed 
orbit  and  8p  normal  to  it.  Then 


(v  +  8v)2  =  (v  + 

\ 

or  to  the  first  order 

d8s    .»  _£  _an  8p    an  8s 

dt  dp  '  v       ds  '  v 

Hence 


d  /Bs\ 

jr  =  —  v^(- 
dt  dt\v  ' 


Again,  let  (u,  u)  be  the  components  of  velocity  in  space  of  P  in  directions 
coinciding  with  8s,  8p.  Since  these  lines  are  rotating  with  the  absolute 
velocity  (0  +  n)  the  kinetic  energy  of  unit  mass  at  P'  is 

dSp  .  )2 

- 


Hence  Lagrange's  equation  for  Sp  is 


Now  this  equation  must  be  satisfied  when  8p  =  8s  =  0,  and  when  the  terms 
which  do  not  vanish  have  been  removed,  it  becomes 


Also  it  must  be  satisfied  when  Sp  =  0,  8s  =  v8t,  where  8t  is  constant,  for  this 
also  represents  a  point  moving  on  the  unvaried  orbit.     Thus 

92F 

-  2  (6  +  n)  v  —  <bv  =  %-—  .  v 
dpds 

and  therefore 

d'&p  .fdSs 

^f-    2(0  +  n 

which  owing  to  (2)  becomes 


214,  215J       The  Restricted  Problem  of  Three  Bodies  245 

But 


, 

Hence  finally 


where 

@  =  w2  +  3(<j> 
a  well-known  result  due  to  Hill. 

Again,  Lagrange's  equation  for  Bs  is 


which  must  be  satisfied  when  Bp  =  Bs  =  0  and  also  when  Bp  =  0,  Bs  =  vBt. 
Hence,  after  removing  the  terms  which  are  independent  of  Bp  and  Bs  and 
then  those  which  contain  Bp, 

fcv  32F 


This  result  may  be  used  to  give  @  another  form,  namely 


v  dt2  "W 

where  V2  =  32/dp2  +  92/9s2  =  92/9£2  +  92/d??2.  This  form  may  be  more  convenient 
than  Hill's  because  V2  (not  to  be  confounded  with  the  three-dimensional  V2) 
does  not  depend  on  any  particular  direction. 

For  some  purposes  it  is  necessary  to  take  the  arc  s  instead  of  t  as  the 
independent  variable.     Then  (3)  becomes 


d  (  dSp\ 

VT-  (V-T*-) 

da  \     ds  J 


or  again,  if  Bp  =  v  ~  *  Bq, 
where 


215.  When  the  unvaried  orbit  is  periodic,  ©  is  a  periodic  function  of  t 
with  the  same  period  T.  The  equation  (3)  is  therefore  a  particular  case  of  a 
linear  differential  equation  with  periodic  coefficients.  Its  general  theory  may 
be  indicated.  Since  the  equation  is  unaltered  when  t  is  replaced  by  1  4-  T, 
g(t  +  T)  is  a  solution  if  g  (t)  is  one.  But  every  solution  is  a  linear  combination 


24(3  The  Restricted  Problem  of  Three  Bodies      [CH.  xix 

of  any  two  others  which  are  independent.     Hence  if  g  represents  g  (t)  and 
G  represents  g(t+T\  glt  g2  being  any  two  solutions, 

#1  =  agi  +  @g2,    G2  =  70i  +  £02 

where  a,  ft,  y,  8  are  constants,  not  unrelated.    For  since  glf  g2  are  two  solutions 
of  (3) 

020i=0>02 

and  therefore 

020i  -  0i02  =  const.  =  G2G,-  G& 

=  (020i  -  0i  0a)  («&  -  £7). 

Hence  «8  —  $7  =  1.     Let/,,/2  be  two  other  independent  solutions.     Then 
0i  =  a/i  +6/J,       02  =  c/i  +  d/2 
G1  =  a^  +  bF2,     G2  =  c 
and  the  result  of  eliminating  g1}  g2,  Gl}  G2  is 


where 

(ad  —  be)  A  =     ada  +  cd/3  —  aby  —  bc8 

(ad  -  be)  B  =  bd  (a  -8)  +  d*/3  -  627 
(ad  -  be)  C  =  -  ac  (a  -8)-  c2/3  -f  a?y 
(ad  —  bc)D  =  —  bca  —  cdft  +  aby  +  adS. 

Hence  A  +D  =  a  +  8  is  a  constant  independent  of  the  choice  of  particular 
solutions,  as  well  as  AD  —  BG  =  a.8  -  fiy  =  1.  But  it  is  now  possible  to  choose 
b/d  and  a/c  so  that  B=C=0.  Then 

F^Afi,     F2  =  Df.2,     AD  =  l 

and  the  functions  /-,,  /2  are  defined  by  the  property  that  they  are  multiplied 
by  constants  when  the  argument  is  increased  by  the  period  T.  Hence  the 
general  solution  of  the  differential  equation  may  be  written 

8p  =  a^e*  <£,  (t)  +  a2e~kt  <£2  (t) 

where  0,,  </>2  are  periodic  functions  with  the  same  period  as  ©  and 
cosh  kT  =  ^  (a.  +  8),  a  constant  which  can  be  derived  from  any  pair  of  inde- 
pendent solutions.  The  quantities  +  k  are  what  Poincare  has  called 
characteristic  exponents.  If  &  is  a  pure  imaginary  circular  functions  are 
involved  and  8p  has  no  tendency  to  increase  beyond  a  certain  limit.  The 
periodic  orbit  is  then  stable.  If  on  the  contrary  k  is  real  or  complex  real 
exponential  functions  are  involved  and  Sp  will  increase  indefinitely.  The 
orbit  is  then  unstable. 

The  question  of  stability  therefore  involves  essentially  the  determination 
of  k.  But  this  is  a  matter  of  great  difficulty  in  general.  What  is  known  as 
Mathieu's  equation,  generally  written  in  the  form 


- 


+  (a  +  16?  cos  2z)  y  =  0 


215,  2ie]       The  Restricted  Problem  of  Three  Bodies  247 

of  which  the  solutions  are  elliptic  cylinder  functions,  is  only  a  particular  case 
of  the  general  type  (3)  and  it  is  the  subject  of  an  extensive  literature.  On 
the  astronomical  side  the  reader  may  consult  Poincare"'s  Methodes  Nouvelles, 
Tome  II.  See  also  Whittaker  and  Watson,  Modern  Analysis,  Ch.  xix. 

216.     The  original  equations  of  motion, 

an  ,    an 


can  also  be  given  a  canonical  form.     Let 


and  then  evidently 


dH  dH 


are  equivalent  to  the  above,  and  they  are  of  the  required  form.     The  integral 
of  energy  is  H  =  0.     Now  consider  the  integral 


/=  [  (-  H  +  Pl%  +  p,-r))  dt. 

Jt0 


Between  fixed  limits  its  variation  will  vanish  along  a  trajectory  in  virtue  of 
the  canonical  equations.  Therefore  it  is  a  minimum  (or  at  least  stationary) 
along  a  trajectory  as  compared  with  its  value  along  any  neighbouring  path. 
Let  the  time  along  any  such  path  be  determined  by  the  equation  of  energy 
H=Q.  Then  the  integral  becomes 


=  f  { 
Jo 


from  which  form,  since  v2  =  2O  —  C,  the  time  is  absent.     Now 

v^d^  +  v^ 
ds  ds 


8  !vds=  r 
J  J0 


f1 
=  l 

Jo 


~        —         ^r 
ds)  V    ds 

dij  »  I1 


248  The  Restricted  Problem  of  Three  Bodies      [CH.  xix 

and 

8  f  l  n  (£dr)  -  r)  df  )  =  n  P 

Jo  Jo 


o 
Therefore,  if  8%  =  ST;  =  0  at  the  limits, 


,-o  i  \   dsj  \  ds, 

Let  the  tangent  to  the  orbit  make  the  angle  <j>  with  the  axis  of  £,  and  let  < 
be  the  normal  distance  to  an  outer  neighbouring  curve,  so  that 

d£  =  ds .  cos  <£,     dr)  =  ds .  sin  <f>,     Sf  =  Sp .  sin  <£,     Brj  =  —  Sp .  cos  0. 
Then 

SJ  =  I    [Svds  —  sin  <f>d  (v  cos  ^>)  8p  +  cos  <f>d(v  sin  0)  Sp  +  2nSpds} 
Jo 

...(5) 


q 

where 

Jr     dv 


—        -- 
op         ds 


J2  being  the  radius  of  curvature.     Along  an  orbit  K  =  0  therefore,  and  this  is 
a  result  already  expressed  in  (1).     It  is  further  to  be  noticed  that 

dK  =  i  azn    /idfi  _i\dv  _  v  dR 

dp  ~  v  dp2      U2  dp      R)  dp  ~  R2  dp 


v  (dp*        v  dp      ~RJ  v  dp 


when  K  =  0,  and  since  v  =  7i(/>  comparison  with  (3)  shows  that 

dK 

@  =  -v^. 
8/> 

It  follows  that  the  action  J  round  a  closed  orbit  is  greater  than  for  any 
adjacent  parallel  curve  when  (H)  is  positive  at  every  point.  In  this  case  the 
periodic  orbit  is  in  general  stable.  Similarly  the  action  J  is  a  real  minimum 
when  ©  is  negative  at  every  point.  Then,  as  (3)  shows,  the  periodic  orbit  is 
obviously  unstable. 

217.  This  remark  is  due  to  Prof.  Whittaker,  who  has  given  another 
application  of  equation  (5).  The  quantity  K  can  be  calculated  for  all  points 
on  a  given  curve.  Now  let  K  be  negative  everywhere  along  a  simple  closed 


216,  2i7]       The  Restricted  Problem  of  Three  Bodies  249 

curve  A.  Then  by  (5)  the  value  of  J  will  be  diminished  when  taken  round 
another  curve  adjacent  to  and  surrounding  A.  Again,  let  the  quantity  K  be 
positive  everywhere  along  another  simple  closed  curve  B  external  to  A.  The 
value  of  J  will  also  be  diminished  when  taken  round  a  curve  adjacent  to  and 
surrounded  by  B.  Now  consider  the  aggregate  of  all  the  simple  closed  curves 
which  can  be  drawn  in  the  ring-shaped  space  bounded  by  A  and  B.  There 
must  exist,  if  the  space  contains  no  singularity  of  Ii,  one  of  these  curves 
which  will  give  a  smaller  value  of  /  than  any  other,  and  it  cannot  coincide 
with  A  or  B  for  any  part  of  its  length.  It  represents  therefore  a  periodic 
orbit  characterized  by  the  constant  of  energy  C,  and  thus  the  existence  of 
such  an  orbit  is  established  when  the  two  curves  A  and  B  can  be  found 
which  satisfy  the  conditions  stated.  The  orbit  is  necessarily  unstable. 

The  same  author  has  given  another  elegant  theorem.    By  Green's  theorem 


(log  V)  d£dr)  =  (Jog  V)dr)-        (log  V 

where  the  first  integral  is  taken  over  the  area  of  a  closed  curve,  and  the  second 
over  its  boundary.     But  if  the  curve  is  a  trajectory,  K  =  0  and  therefore 

3    .  d<b      2n 

0  =  —  (logt;)  +  -^  +  - 
dp  as       v 

9/i        \  9f      3/1        ,drj     dd>      2n 

=  ^  (log  v)  ^  +  5-  (log  v)  a  '  +  j    +  ~ 
9£  op     dt)  op      ds       v 


3  drj      3    .        .d£d(f>2n 

=  ^  (log  v}  -T-  -  5~  0°g  V)  J     +   j     +  — 

d  ds      di  ds      ds       v 


Hence 


=  -  l 


This  assumes  that  the  enclosed  area  contains  no  singularity  of  the  integrand. 
But  this  function  becomes  infinite  at  the  centres  of  attraction.  Surround  the 
mass  p  at  (—  c, ,  0)  with  a  small  circle  K^  of  radius  p.  Then  since 

the  integral  round  the  circumference  becomes 

a      <.  a\ ,  r  i  /7  a      ,.3 


=  —  7T. 


250  The  Restricted  Problem  of  Three  Bodies      [OH.  xix 

Similarly  the  corresponding  integral  round  a  small  circle  K«  surrounding  the 
mass  v  tends  to  the  same  limit.  Now  if  the  outer  boundary  contains  either 
of  the  attracting  masses  or  both,  the  boundary  integral  must  be  diminished 
by  subtracting  the  integrals  taken  round  ^  or  /c2  as  the  case  may  be.  Hence 
the  final  result  is 

8^      i 
JVd£2     c 

where  j=  0,  1  or  2  according  as  the  loop  of  the  orbit  contains  neither  or  one 
or  both  of  the  attracting  masses,  j  is  the  total  angle  through  which  the 
tangent  to  the  orbit  turns,  and  T  is  the  time  from  one  end  of  the  loop  to  the 
other.  In  the  case  of  a  periodic  orbit  in  the  form  of  a  single  closed  curve 

7=  27T. 

218.  The  equations  of  relative  motion  are  capable  of  a  transformation 
which  is  very  useful  in  some  cases.  This  may  be  deduced  from  the  intro- 
duction of  conjugate  functions  in  a  general  form.  Let  the  original  equations  be 


dV 
77  +  2w£  —  n2rj  =  — 

Of) 


or  in  the  Lagrangian  form 

_d  f_ 

dt  Va£y  "  af 

d  (dT\_<W  =  dV 
<ft\39/     ?T?  ~  817 

where 

^=i(i-^)2  +  i(^  +  o2 

and  the  integral  of  energy  is 

Hf2+^)  =  ^2(£2  +  7f)+F- 
Now  let 

£  +  "7  =f(u  +  tv)t      t2  =  -  1 
so  that 

5J:  _  dij       dg  _     drj 
du     dv  '     dv         du 
and 

d       .  d       .3 

T*  =  u^~  +  v^~- 
at        ou        dv 

Also  let 


d(u,  v)     du'  dv      dv'du 
Then  if 


217,  2is]       The  Restricted  Problem  of  Three  Bodies  251 

where  the  suffix  denotes  the  degree  of  the  terms  in  u,  v  (or  £,  ??),  it  will  be 
found  that 


du     *  duj  V      '  dv 


The  equations  of  motion  may  now  be  written 


_ 

dt\duj      dt\du         du       du       du       du 

A  (dll\4.  d  ^_8_77l  =  ?^  +  ??«  +  — 
dt  \dv)      dt\dv)       dv       dv       dv       dv 

and  the  integral  of  energy  is 

T2=T»  +  V-h. 

It  can  be  verified  without  difficulty  that 


Also 


d      /a-/l\  dJ-  1  01          r. 

TI  ( -5T  I  —  -3T  ™  •"  2fl«'*' 

a^  \duj       du 


0  _  0 

du  +  du  +  du  ~  2  du  ^  }      du  *  du 


Hence  the  equations  of  motion  become 


Now  let 

dt  =  JdT,    ^ 
and  we  have 

dzu       T  dv  _  air 

dT^~      JdT~du 


,  du  _  an' 

JdT~~dv 


with  the  equation  of  energy 


(dTj  +dTj 


252  The  Restricted  Problem  of  Three  Bodies      [CH.  xix 

It  is  convenient  to  write 

/i  =f(u  +  tv),    /2  =f(u  -  iv), 
and  then 

\du 

219.  What  is  needed  when  F  is  the  potential  due  to  two  masses  p,,  v  at 
a  distance  2c  apart  is  a  transformation  of  the  coordinates  which  will  rationalize 
both  the  distances  plf  p.2.  Such  a  transformation  is 

£  -f  irj  =  b  +  c  cos  (u  +  iv),     b  —  c(/n  —  v)/(/J>  +  v) 

where  b  is  the  distance  of  the  middle  point  between  the  masses  from  their 
centre  of  gravity.     For 

pi*  =  (£  —  b  +  c)2  +  rf  =  4c2  cos2  -|  (u  +  iv)  cos2  \  (u  —  iv) 
PZ  =  (£  —  b  —  c)2  +  if  —  4c2  sin2  ^  (u  +  iv)  sin2  ^(u  —  iv) 
and  hence 

,r       U,         V  U>  V 

PI     PZ     c  (cosh  v  +  cos  u)      c  (cosh  v  —  cos  u) ' 
Also 

/=////  =  c2  sin  (u  +  iv)  sin  (u  —  iv)  =  |  c2  (cosh  2v  —  cos  2a) 
and 

£2  +  if  =/i/a  =  b2  +  26c  cosh  v  cos  u  +  ^c2  (cosh  2w  +  cos  2w). 
Hence 

H'  =  /*c  (cosh  v  —  cos  u)  +  vc  (cosh  v  +  cos  w) 

+  %n2bc3  (cosh  3-y  cos  u  —  cosh  v  cos  3u)  +  y^w2c4  (cosh  4v  —  cos  4u) 
—  ^c2  (h  —  £w262)  (cosh  2w  —  cos  2u) 
and  the  equations  of  motion  are 

,  dv      9H' 


-^  +  nc*  (cosh  2v  -  cos  2u)  ^  =  -^- . 
The  time  is  given  by  a  final  integration 

$  =  £c2 1  (cosh  2v  -  cos  2w)  rfT=  IfrpidT. 

These  equations  are  in  general  very  complicated,  although   they  offer 
essential  advantages  in  studying  the  motion  in  the  immediate  vicinity  of 


218,  219]       The  Restricted  Problem  of  Three  Bodies  253 

one  of  the  masses.     Two  particular  cases  may  be  noticed.     In  the  first  the 
masses  are  equal,  /JL  =  v  and  6  =  0.     The  equations  of  motion  then  become 

d?u  dv 

-T     -  nc-  (cosh  2v  —  cos  2u)  -     =  —  <?h  sin  2u  +  |  w2c4  sin  4w 


__  .f  nc*  (cosh  2v  —  cos  2u)  -^  =  2/zc  sinh  v  —  c2/i  sinh  2v  +  |  ft2c4  sinh  4w 

Ct7  "  (IJ- 

which  are  equivalent  to  equations  given  by  Thiele  and  employed  by  Stromgren 
and  Burrau.  The  other  case  represents  the  problem  of  two  centres  of  attrac- 
tion fixed  in  space,  so  that  n  =  0.  Then  the  equations  become  simply 

d?u      .          ....  .  ,,    •    0 

-T™^  =  (p  —  v)  c  sin  u  —  c2h  sin  2u 

2 

j,  +  vc  sinh  v  —  c2h  sinh  2t>. 


—  —        , 

Here  the  variables  u,  v  are  separated  and  the  equations  lead  immediately  to 
a  solution  in  elliptic  functions.  The  comparison  of  this  problem  with  the 
simplest  case  of  the  problem  of  three  bodies  is  instructive  as  to  the  difficulty 
of  the  latter. 


LUNAR    THEORY    I 

220.  The  theory  of  the  Moon's  motion  relative  to  the  Earth  has  been 
discussed  with  generally  increasing  elaboration  and  completeness  by  various 
authors  from  the  time  of  Newton  to  the  present  day.     The  methods  which 
have  been  employed  also  differ  considerably,  presenting  peculiar  advantages 
in  different  respects,  so  that  it  cannot  be  said  definitely  that  any  one  method 
possesses  an  exclusive  claim  to  consideration.     But  at  the  present  time  three 
modes  of  treatment  are  certainly  of  outstanding  importance,  those  adopted 
by  Hansen,  Delaunay  and  G.  W.  Hill  respectively.     Hansen's  theory  was 
reduced  to  the  form  of  tables  by  the  author  ;  these  tables  were  published  in 
1857  and  are  still  in  common  use,  but  will  shortly  be  superseded.    Delaunay  's 
work  took  the  form  of  an  entirely  algebraic  development  of  the  Moon's  motion 
as  conditioned  by  the  Earth  and  Sun  alone.     His  theory  has  been  completed 
by  others  and  made  the  basis  of  tables  recently  published.     Hill's  researches, 
which  bear  a  certain  relation  to  Euler's  memoir  of  1772,  deal  only  with 
particular  parts  of  the  theory,  but  the  whole  work  on  these  lines  has  now 
been  carried  out  systematically  and  completely  by  E.  W.  Brown  and  will 
form  the  foundation  of  a  new  set  of  lunar  tables  now  in  course  of  preparation. 

Here  it  is  only  possible  to  attempt  a  slight  sketch  of  one  method.  For 
this  purpose  Hill's  theory  will  be  chosen,  partly  because  it  is  destined  to 
receive  extensive  practical  application,  and  partly  because  it  contains  original 
features  of  the  greatest  theoretical  interest.  The  reader  who  wishes  to  gain 
a  comparative  view  of  the  different  methods  which  have  been  used  in  the 
lunar  theory  will  study  Brown's  Lunar  Theory  and  may  also  be  referred  to 
the  third  volume  of  Tisserand's  Mecanique  Celeste. 

221.  Let  the  mass  of  the  Earth  be  E,  of  the  Moon  M  and  of  the  Sun  m  , 
the  unit  being  such  that  the  gravitational  constant  G  =  1.     Let  the  origin  of 
rectangular  axes  be  E,  (x,  y,  z)  the  coordinates  of  M  and  (x  ,  y',  z'}  the  co- 
ordinates of  m.     Further,  let  r  be  the  distance  EM,  r'  the  distance  Em, 
and  A  the  distance  Mm'.     Then  (§  23)  the  forces  on  the  Moon  per  unit  mass 
relative  to  E  can  be  derived  from  the  force  function 


r,  M     m      m  ,     . 

F=  -—  +  -  -  -    (xx'  +  yy'  +  zz'} 


220,  221]  Lunar  Theory  I  255 

by  differentiation  with  respect  to  x,  y,  z ;  and  similarly  the  forces  on  the  Sun 
per  unit  mass  relative  to  E  can  be  derived  from  the  function 

E+m      M     M.     , 

F  = -, h  -i 7  (xx  +yy  +  zz) 

r  A      r3 

by  differentiation  with  respect  to  x',  y',  z'.     Hence  the  ^-component  of  the 
Sun's  acceleration  relative  to  G,  the  centre  of  gravity  of  E  and  M,  is 

r)  W  M          rl  T?  1''  T'  —  V  V, 

u  r  J.u        oju  ,  -r-,          /,   i//         i  g  i//        ***        H  r  ** 

I  Lj   '  I  A™    '     \  /l/f ^_  lift 

— rrr  —  I  JTJ  ^-  ft  I,    t  — —   —  jifj.    r ^^  ML    — 

fjy  rlj     \     /L/    H  np  /  f\  7^ 

?L/  (  rv*  rr*  —   f  flf* 

+  .,      ,.;  \(E  +  M}  -. -  +m'-      -+mf  — 
E  +  M  \  r*  A3  r- 

E+M+m   (»x_      ^x'-x 

/Q       ' 


A« 

This  expression  will  be  derived  by  differentiating  the  function 


M+m'  (E     M\ 
Vr/  +  Ay 


with  respect  to  a;',  or  with  respect  to  xl,  where  (xlt  y1(  z^)  are  the  new  co- 
ordinates of  m'  when  parallel  axes  are  taken  through  G  instead  of  E.  Let  rl 
be  the  distance  m'G,  6^  the  angle  m'GM  and  8=  cos  0,.  Then 


r'-l  =  \r^  +  -^—^rrrlS  + 


and 


(     M       r  M2        r2  ) 

=  rri  1  _   j^n  r-  P!  +  7Jj=^jy  -*  A  -  •-  [ 


2#  #2 

A  —1   )  «  2  _    ««    .Cf     i 

L^  ~t'l  T^.       •>r'/l^-'     i^    /   ».?     .        i 


E       r  E2        r2 

1    -L  P    4-  P     4- 

f^T^n     '     (^  +  !/)2r12jr2+-". 


where  P1}  P2,  ...  are  Legendre's  polynomials 

Hence,  when  expanded  in  terms  of  r/r1} 

E  +  M+m'  L          EM 


Now  the  Moon's  parallax  is  of  the  order  1  °,  the  solar  parallax  is  of  the 
order  9"  and  the  ratio  M/E  is  of  the  order  1/80.  It  follows  that  the  second 
term  in  F^  is  of  the  order  10~7  as  compared  with  the  first.  It  can  be 
neglected,  at  least  in  the  first  instance.  F^  is  therefore  reduced  simply  to 
the  first  term,  and  the  meaning  of  this  is  that  the  motion  of  G  about  m',  or 
of  m'  about  G,  is  the  same  as  if  the  masses  E  arid  M  were  united  at  their 
centre  of  gravity. 


256  Lunar  Theory  I  [OH.  xx 

This  motion  is  elliptic  and  the  coordinates  (x1}  y1}  z±}  can  be  treated  as 
known  functions  of  the  time  according  to  undisturbed  elliptic  motion.  The 
influence  of  the  other  planets  is  left  out  of  account  in  the  first  instance  and 
finally  introduced  in  the  form  of  small  corrections.  The  first  task,  and  the 
only  one  considered  here,  is  to  find  an  appropriate  solution  of  the  problem  of 
three  bodies,  the  problem  being  already  so  far  simplified  that  the  relative 
motion  of  the  Sun  and  the  centre  of  gravity  of  the  Earth-Moon  system  is 
supposed  known. 

222.  The  force  function  F  is  expressed  in  terms  of  (x,  y',  z')  and  not  the 
coordinates  (xl}  yly  z-^)  now  supposed  known.  It  is  necessary  to  consider  the 
effect  of  this.  The  ^-component  of  the  Moon's  acceleration  is 

dF  ...  x         ,x  —  x         ,  x 

=-  =  -  (E  +  M)  -  -  in  —  —  -  m  — 
das  r*  A3  r3 

x      mf     E  \      m!  (    M 

)~~~X  *     fr*  ' 


snce 

x'  =  ^  +  Mxj(E  +  M\    x-x'  =  -xl  +  ExK  E  +  M). 

This  component  is  clearly  derivable  from  the  force  function 

E  +  M     m'(E  +  M)     m'  (E  +  M) 
1-  ~V~          ~^£A~  Mr' 

when  r   and  A  are  expressed  in  terms  of  (xlt  y1}  z-^)  instead  of  (x  ',  y',  z'}. 
When  A"1,  r'~l  are  expanded  in  terms  of  rjr^  this  becomes 


^E+M     m'  \(E+My>      r^  E2  -  M  2  r3  E3  +  M3  r4 

r       f  r,  |     EM      +  r,-       +  (E  +  Mf  ^    3  +  (E  +  M)3  ^     4 


E+M     m'r*  (          E-M  r_          E* 
~~     '~  *  3+  * 


for  the  term  in  1/rj  does  not  contain  {x,  y,  z)  and  can  therefore  be  suppressed. 

As  a  matter  of  fact  the  force  function  which  is  commonly  used  for  the 
motion  of  the  Moon  is  neither  F1  nor  the  function 

„  _  E  +  M     m?  _  m'r       fi 
r  A       r'2 

where  0  is  the  angle  m'EM,  but  the  function 

_      E+M     mf     m'r  ~ 

which  is  derived  from  F  by  substituting  the  coordinates  of  the  Sun  relative 
to  G  for  the  coordinates  relative  to  E.     Thus 


221-223]  Lunar  Theory  I  257 

and  therefore  in  the  expanded  form 


after  suppressing  m  jrl.  This  is  not  the  same  as  Flt  but-  for  practical 
purposes  it  can  be  brought  into  agreement  by  a  simple  device.  Let  a,  a'  be 
the  mean  values  of  r,  ?v  It  is  found  that  to  a  term  of  the  series  involving 
(r/r^  correspond  inequalities  with  the  factor  (a/ay.  If  then 

(E  -  M)  a/(E  +  M)  a' 

be  substituted  for  a/a'  in  the  results  which  follow  from  the  use  of  F2,  they 
will  be  very  nearly  the  same  as  if  they  had  been  derived  by  using  Fl.  It 
may  be  left  to  the  reader  to  examine  the  order  of  the  chief  outstanding  dis- 
crepancy after  this  treatment  of  F^  It  is  easy  to  make  the  adjustment  exact. 

223.  Let  the  axis  Ez  be  taken  normal  to  the  ecliptic  and  let  EX,  EY 
rotate  in  the  ecliptic  plane  of  (xy)  with  the  Sun's  mean  motion  n'.  The 
equations  of  motion  of  the  Moon  are  then 


Now  if  E  +  M  =  /A,  since  n/sa'8  =  m'  (more  strictly  ra'  + 

'-  -  (f  r*S* 

*   A    ^6 


the  higher  terms  containing  r/rj  and  therefore  the  solar  parallax  as  a  factor. 
Let  v'  be  the  true  longitude  of  the  Sun  and  let  v'  =  e  when  t  —  0.  Then  the 
Sun's  coordinates  are 

X'  =  T!  cos  (v'  —  n't  -  e'),      Y'  =  ra  sin  (V  —  w'£  —  e'),     /  =  0 

the  axis  of  X  being  always  directed  towards  the  Sun's  mean  place.  When 
the  solar  eccentricity  is  neglected  and  the  Sun's  orbit  treated  as  circular, 
v'  —  n't  +  e'  and  TJ  =  a,  so  that 

X'  =  r,  =  a\     Y'  =  z'  =  0,     rS  =  (XX'  +  YY')/^  =  X. 
Hence  when  the  solar  parallax  and  eccentricity  are  both  neglected 
F2  =  pr-1  +  ri*  (f  X*  -  £r2)  =  yu?-1  +  n'2  (X2  -  $  72  -  \z*) 

P.  D.  A.  17 


258  Lunar  Theory  I  [CH.  xx 

and  when,  still  further,  the  latitude  of  the  Moon  is  ignored,  the  equations 
of  motion  become  simply 

X  -  Zn'Y-  3?i/2X  =  - 


These  two-dimensional  equations  represent  the  simplest  problem  bearing  any 
real  resemblance  to  the  actual  circumstances  of  the  lunar  theory.  It  is  the 
degenerate  case  of  the  restricted  problem  of  three  bodies  when  the  two 
finite  masses  are  relatively  at  a  very  great  distance  apart  and  refers  strictly 
to  the  motion  of  a  satellite  in  the  immediate  neighbourhood  of  its  primary. 
These  equations  have  great  importance  in  Hill's  theory. 

Again,  when  the  solar  parallax  alone  is  neglected,  F2  may  be  written  in 
the  form 


where  the  third  term,  which  vanishes  with  the  solar  eccentricity,  is  a  quadratic 
function  in  X,  Y,  z.     Thus 

* 

where  A',  H',  B',  G'  are  functions  of  t  to  be  derived  from  the  elliptic  motion 
of  the  Sun.     The  equations  of  motion  now  become 

•*r  fy       I  \T  Q        /«  T7"       j  A    f  ~\T        -          TTf  T7 

17    i    O^.'  V  i     ZT'  V    i     Tf'  V 

i  +  Zn  A  +  li  A  -f  Li  I  — 

z  +  n'2z     +C'z 

and  these  are  the  foundation  of  the  researches  of  Adams  into  the  principal 
part  of  the  motion  of  the  lunar  node. 

224.     It  is  now  necessary  to  give  Hill's  transformation  of  the  general 
equations  of  motion.     Let 

n  a 

m  =        — , ,      K  =  - —    -       ,     v  =  n  -  n . 
n  —  n  (n  —  n  )- 

Then,  since  r2  =  us  +  z2,  n  being  undefined  as  yet, 

a'3 
2m2  — (P2r2  +  P3r3/r1+  ...) 

M3X^  3/1*  / 


where  H2',  Q3,  ...  are  homogeneous  functions  in  u,  s,  z  of  degree  2,  3,  ...  and 
of  degree  0,  -  1,  ...  in  a'.     Let  £1'  =  fla'  +  O3  +  .  .  .  . 

The  kinetic  energy  of  the  Moon  T  is  given  by 


=  (u  +  ii'iu)  (s  —  n'is)  +  z2. 


223-225]  Lunar  Theory  I  259 

The  equations  of  motion  are  therefore 

O  ET 

ii  +  2niii  —  n'2u  =  2  — 2 
ds 

s  —  2n'is  —  n'2s  =  2  ~ 
du 

Let 

where  t0,  like  n,  is  a  constant  at  present  undefined.     The  previous  equations 
become 

3D' 

D*u  +  2mDu  +  mht,  =  Ku/^  -     — 

ds 

D2s  -  2mDs  +  irfs  =  KS/rs  -     — 

du 


--. 

"  dz 

It  is,  however,  convenient  to  separate  from  fl/  (accented  for  this  reason)  the 
part  which  is  independent  of  the  solar  eccentricity.     This  is 

IV  -  I2a  =  m2  (3Z2  -  r2)  =  | in2  (a  +  s)2  -  m2  (us  +  £2). 
With  this  change  the  equations  of  motion  take  the  form 


Dhi  +2mDu  +  f  m3  (u  +  s)-  —  =  -      ^ 

r3  ds 

IPs  -2mDs  +  4m2  (u  +  s)--  =-      — 


r3  "  dz 

where  ft  =  fla  +  H8  +  . . . .     Thus 


•(2) 


-m^-l     ............  (3) 

;  \»i          / 

which  vanishes  with  the  solar  eccentricity. 

225.     The  next  object  is  to  transform  the  equations  in  u  and  s  so  as  to 
remove  the  terms  involving  r~s.     Since  (§  123) 


and  ^2  contains  terms  involving  £  explicitly  only  in  H,  in  this  case 


ws  +  z*  -  n'*us  =  2,P2  -^- 

ot 


17—2 


260  Lunar  Theory  I  [CH.  xx 

or  in  the  later  notation 

Du .  Ds  +  (Dzf  +  |  m2  (u  +  s)2  -  mW  +  ^  =  0  -  ft  +  -D 


r 

where  C  is  a  constant  of  integration,  D~l  is  the  inverse  operator  to  D,  and  Dt 
represents  the  operator  D  applying  to  ft  only  in  so  far  as  ft  contains  t 
explicitly.  This  corresponds  to  the  equation  of  energy. 

Again,  since  r2  =  us  +  z-,  the  equations  of  motion  (2)  give 
sD2u  +  uD2s  +  2zD2z  +  2m  (sDu  -  uDs)  +  f  m2  (u  +  s)2  -  2m2z2  -  2 

_/  8_ft       8_ft       8_ft; 
\    ds         du         dz 

by  Euler's  theorem,  ftp  being  a  homogeneous  function  of  degree  p  in  u,  s,  z. 
The  result  of  adding  the  last  two  equations  is 

D2  (us  +  z2)  -Du.Ds-  (Dz)2  +  2m  (sDu  -  uDs)  +  f  m2  (u  +  s)2  -  3m2  ^ 

.(4) 


This  is  one  equation  of  the  required  form. 

The  other  equations  are  obtained  simply  by  eliminating  the  terms  with 
r~3  as  a  factor  between  different  pairs  of  the  equations  of  motion.  Thus 
from  the  first  pair 

D  (uDs-sDu  -  2mus)  +  f  m2  (u2  -  s-2)  =  s8-  -  u  ^  ...(5) 

ds          du 

and  when  the  third  equation  is  used, 

8f  1  r)fi 

D  (uDz  —  zDu)  —  2mzDu  -  im2^  (ou  +  3s)  =  z lu  -= 

ds  dz 

D  (sDz  -  zDs)  +  2mzDs  -  \\tfz  (3w  +  5s)  =  z'd~  -  Is  ^ 

Oil  OZ 

These  combined  give 

D  {(u  ±  s)  Dz  -zD(u±  s)}  -  2mzD  (u  +  s)  -  m2zW 

an    8n\  ,8n 

*-  ±  3-  -  HM  ± s)  o- 

8s       du/  dz 

where  with  the  upper  sign  W  =  4<(a  +  s)  and  with  the  lower  W  =  u  —  s.  In 
this  more  symmetrical  form  the  real  and  imaginary  parts  of  u  and  s  are 
clearly  separated. 

Equations  in  the  form  of  (4)  and  (5)  have  two  advantages.  In  the  first 
place  the  left-hand  members  are  homogeneous  in  u,  s,  z  of  the  second  degree. 
Except  for  the  constant  C  this  applies  also  to  the  right-hand  members  when 
the  parallax  of  the  Sun  is  neglected,  and  the  parallactic  terms  need  rarely  be 
taken  beyond  the  third  and  fourth  degrees.  In  the  second  place,  whereas 
X  and  F  can  be  expressed  as  trigonometrical  series  in  terms  of  t,  u  and  s 
can  be  expressed  as  algebraic  (Laurent)  series  in  terms  of  £  and  such  series 


225,  226]  Lunar  Theory  I  261 

can  be  more  easily  manipulated.     Also  if  u  =/(£)•  s  =/(£-»)  and  therefore 
when  either  u  or  s  has  been  calculated  the  other  can  be  derived  immediately. 

226.  The  general  method  of  the  lunar  theory,  which  is  common  to  all 
forms,  consists  in  choosing  an  intermediate  orbit  which  bears  some  re- 
semblance to  the  actual  path  of  the  Moon  and  in  studying  the  variations 
which  it  must  undergo  in  order  that  the  path  may  be  represented  accurately 
and  permanently.  This  intermediate  orbit,  since  it  merely  serves  as  a  subject 
for  amendment,  will  naturally  be  chosen  with  a  view  to  simplicity.  At  the 
same  time,  the  more  closely  it  represents  the  permanent  features  of  the 
actual  motion,  the  less  burden  will  be  thrown  on  the  subsequent  variations. 
Thus  one  might  take  the  osculating  elliptic  orbit  of  the  Moon  about  the 
Earth  as  the  intermediary,  neglecting  the  effect  of  the  Sun  altogether.  The 
intermediate  orbit  adopted  by  Hill  is  called  the  variational  curve  and  this 
must  now  be  defined. 

When  the  solar  eccentricity  (<?')  and  the  solar  parallax  are  neglected, 
fit  =  0.  Also,  when  the  Moon's  latitude  is  neglected,  z  =  0.  Equations  (4) 
and  (5)  then  become 

D* (us)  -  Da .  Ds  +  2m  (sDu  -  uDs)  +  f  ma  (u  +  s)2  =  0\ 


which  must  be  equivalent  to  (1),  whence  in  fact  they  can  be  directly  deduced. 
The  constant  K  (or  /LI)  has  been  eliminated  and  the  constant  C  has  been 
introduced.  There  must  be  a  relation  between  them  which  can  be  found  by 
reference  to  the  original  equations  of  motion.  Hill's  variational  curve  is 
defined  as  that  particular  solution  of  (1)  or  (6)  which  represents  a  periodic 
orbit.  Since  the  axes  of  reference  rotate  at  the  rate  n'  the  period  of  this 
orbit  must  be  2Tr/(n  —  n')  where  n  is  the  mean  motion  of  the  Moon.  From 
this  it  follows  that  the  coordinates  X,  Y  of  the  solution  have  this  period  and 
can  be  expressed  in  the  form  of  Fourier  series  in  (n  —  n'}  t,  while  u,  s  can 
be  expressed  in  the  form  of  Laurent  series  in  £  The  coefficients  will  be 
developed  in  powers  of  m,  and  this  is  an  essential  advantage  of  the  method, 
since  it  is  precisely  this  development  which  is  less  easy  by  the  earlier 
methods.  As  a  particular  solution  of  the  equations  the  symmetrical  periodic 
orbit  involves  no  arbitrary  constants  beyond  those  already  introduced,  namely 
n,  which  de'pends  on  the  actual  scale  of  the  lunar  orbit,  and  t0,  which  gives 
an  arbitrary  epoch  corresponding  with  the  fact  that  (6)  do  not  involve  the 
independent  variable  explicitly. 

The  existence  of  such  periodic  orbits  is  assumed.  The  question  has  been 
discussed  analytically  by  Poincare  (Methodes  Nouvelles,  Tome  i),  who  has 
proved  that  they  do  exist  in  general.  To  some  extent  the  assumption  will 
be  found  practically  justified  by  the  results.  But  there  is  no  doubt  on  the 
point.  The  periodic  orbit  in  the  actual  circumstances  could  be  found  by  the 
method  of  quadratures. 


262  Lunar  Theory  I  [OH.  xx 

227.  The  assumption  that  the  periodic  orbit  required  is  symmetrical 
about  both  axes  at  once  limits  the  form  of  the  expansions.  For  with  this 
limitation  X,  Y  must  be  of  the  form 

X  =  2  A*+1  cos  (2i  +  1)  £,     F=  2  A'^  sin  (2f  +  1)  £     £  =  (n  -  n')  (i  -  *„) 
o  o 

where  Y=  0  when  £  =  t0.     Hence 

u  =  2  ft  (A*»  +  A'*H)  £*»  +  $  (^+1  -  ^+1)  £--->}  =  a  2  a*  f*« 

0  -oo 

0  =  I  ft  (4*H  -  ^WO  £*+1  +  1  (^+1  +  ^i  WO  f-*-1!  =  a  I  a-^  £2i+1 

0  -so 

where 


-"-21+1  =  3-  (&2i  ~t~  &—  2i—  a)>       •"•  at+i  =  a  (a2i  ~~  a-2i—  2)- 

As  it  is  necessary  to  multiply  such  series  together  and  to  exhibit  the  products 
as  double  summations,  it  is  convenient  to  write 


or  similar  equivalent  forms,  so  as  to  retain  always  a  fixed  coefficient  a2i  and  a 
fixed  power  go  in  the  typical  constituent.  The  result  of  substituting  the 
series  in  (6)  is  : 

ar*C  =  22  47*0*  a^j+zi  &  -  22  (2i  +  1)  (2j  -  2t  -  1)  a* 


+  2m 

+  f  m2  2  2  a^-  (2a-$ 
»  j 

0  =  2  2  2j  (2j  -  4t  -  2)  a2l-  a_2j+2i  ^  -  2m  22  2ja2i  a 

1   3 

+  f  m2  2  2  a*  (a^-af^  -  a_2?-_2i_2)  f2-' 


»  j 

where  *  and  j  have  all  positive  and  negative  integral  values.    The  coefficients 
of  every  power  of  f  must  vanish  identically,  and  therefore 

a~2C  =  2  {(2t  +  I)2  +  4m  (2i  +  1)  +  f  m2}  a2^  +  f  m2  2^  a-2t-2  .  .  .(8) 

*  i 

when  J  =  0,  and 

0  =  2  {4J*  +  (2i  +  1)  (2i  +  1  -  2J)  +  4m  (2t  +  1  -  j)  +  $  m2} 
+  f  m2  2  Oaf  (astf_8i_a  +  a_2j_2(-_2) 

i 

0  =  -  2  4j  (2i  +  l-j  +  m)  a2i  a_,?-+2(;  +  fma  2  a2l-  («2j-2i-2 

*  i 

whenj/  has  any  other  value. 


227-229]  Lunar  Theory  I  263 

228.  Owing  to  the  introduction  of  a,  one  coefficient  «0  may  be  made 
equal  to  1,  though  retained  for  the  sake  of  symmetry.  Then,  if  m  is  a 
small  quantity  of  the  first  order,  ap  is  found  to  be  of  order  \p  ,  being  a 
function  of  m  alone.  This  fact  makes  it  possible  to  obtain  the  coefficients 
by  a  process  of  continued  approximation,  provided  m  is  sufficiently  small. 
The  terms  containing  a0a2j,  a0a_2j  in  the  last  equations  are  obtained  when 
i  =j  and  i  =  0,  and  they  are  respectively 

|4j2  +  2j  +  l  +  4m  (  j  +  1)  +  |m2}  a0«2j  +  {4j2  -  2j  +  I  -  4m  (  j  -  1)  +  f  m2}  a0a_2j 

and 

-  4j  (1  +  j  +  m)  a0a2j  -  4j  (1  -  j  +  m)  a0a_2j  ..............  .(9) 

Let  the  two  equations  be  combined  so  as  to  eliminate  the  second  of  these 
terms.     The  result  may  be  written  : 

2  a2i  |[2j,  2t]  a_y+2i  +  [2j,  +]  a^_2l-_2  +  [2j,  -]  a_2j-_2i-2}  =  0  ...(10) 

i 

where 

F9  '  9  H  -  _  i    fy'2-2-4m  +  m3  +  4  (t  -  j)  Q'-l-m) 
j  '  '~~8f-  2  -  4m  +  m2 

3m2    4j2-8j-2-4m(j  +  2)-9m2 

W*  4         "  T6f  '  "          8f  -  2  -  4m  +  m2 

3m2    20j»  -  16j  +  2  -  4m  (5j  -  2)  +  9m2 
16j2  '  8j2  -  2  -  4m  +  m2 

the  common  divisor  being  chosen  so  that  the  coefficient  of  a0a2j,  [2J,  2j], 
is  -  1,  while  [2j,  0]  =  0. 

If,  on  the  other  hand,  the  term  in  a0a2j  be  eliminated,  the  result  will  be 
found  to  be 

f-a  =  0 


which  can  be  deduced  from  the  same  series  of  equations  (10)  by  changing 
the  sign  of  j  and  then  writing  i  —j  for  i  in  the  first  term.  This  single  series 
is  therefore  sufficient.  The  last  equation  can  also  be  written 

2  {[-  2J>  ~  2*1  a2j-2itt-2i  +  [-  2j,  -]  a2j_2i_2a2i  +  [-  2j,  +]  a_2;-_2i_2a2i}  =  0 
i 

and  hence  the  rule  for  connecting  the  pair  of  equations  corresponding  to  ±j: 
in  terms  multiplied  by  [2j,  2i]  change  the  signs  of  j  and  i  throughout  (both 
in  coefficients  and  in  suffixes)  ;  in  the  other  terms  write  [—  2J,  —  ]  for  [2j,  +] 
and  [—  2j,  +]  for  [2j,  —  ],  the  suffixes  being  unchanged. 

229.    Since  the  coefficients  [2j,  +]  are  of  the  second  order  in  m,  the  orders 
of  the  three  terms  are  respectively 

2  i  +2  %  -  j  ,     2  i  +2   i+l-jl  +  2,     2  |  i  \  +  2   i  +  l+j  +2 
which  are  at  least 


264  Lunar  Theory  I  [CH.  xx 

Let  the  equations  be  written  down  so  as  to  include  all  quantities  of  the  sixth 
order  (neglecting  m8).  This  requires  j  =  ±  1,  ±2,  ±3.  The  orders  of  the 
terms  with  the  only  possible  values  of  i  are : 

j  =  1,     i  =  2  (6,  10,  14),  1  (2,  6,  10),  0  (2,  2,  6),  -  1  (6,  6,  6),  -  2  (10,  10,  6) 

j  =  2,     t  =  2  (4,  8,  16),  1  (4,  4,  12),  0  (4,  4,  8) 

j  =  3,     i  =  3  (6,  10,  22),  2  (6,  6,  18),  1  (6,  6,  14),  0  (6,  6,  10). 

Hence  the  required  equations  are : 

a002   =  [2,  4]  a2a4  +  [2,  -  2]  a_2a_4  +  [2,  +]  (2a2a_2  -f  a02)  +  [2,  -]  (2a0a_4  +  a2_2) 

a0a_2  =  [-  2,  -  4]  a_2a_4  +  [-  2,  2]  «2a4  +  [-  2,  -]  (2a2  a_2  +  a02) 

+  [-2,  +](2a0a_4  +  a2_2) 
a0a4   =  [4,  2]  a2a_2  +  [4,  +]  2a0«2  • 
a0a_4  =  [-  4,  -  2]  a2a_2  +  [-  4,  -]  2a0a2 
«0a6   =  [6,  4]  a_2a4  +  [6,  2]  a2a_4  +  [6,  +]  (2a0a4  +  a*,2) 
a0a_6=  [-  6,  -  4]  a2a_4  +  [-  6,  -  2]  a_2a4  +  [-  6,  -]  (2a0a4  +  a22). 
Thus,  since  a0  =  1,  if  m6  be  neglected, 

a2=[2,  +],     a_2  =  [-2,  -] 
and  then,  neglecting  m8, 

«4    =  [4,  2]  [2,  +]  [-  2,  -]  +  2  [4,  +]  [2,  +] 
a_4  =  [-  4,  -  2]  [2,  +]  [-  2,  -]  +  2  [-  4,  -]  [2,  +]. 

These  values  will  give  a6,  a_6  as  far  as  m9,  and  inserted  on  the  right-hand 
side  of  the  first  pair  of  equations  they  give  second  approximations  to  «2,  a_2 
of  the  same  order.  It  is  to  be  noticed  that  each  stage  of  further  develop- 
ment carries  an  equation  four  orders  higher. 

The  ratio  of  the  mean  motions  of  the  Sun  and  Moon,  and  therefore  the 
numerical  value  of  m,  is  known  with  great  accuracy  from  observation.  Hill 
adopted  the  value 

m  =  n'/(n  _  w')  =  0'08084  89338  08312. 

Hence  it  is  practicable  to  introduce  the  numerical  value  of  m  from  the 
beginning,  and  the  approximation  to  great  accuracy  in  the  calculation  of 
a*2,  ...  is  then  extremely  rapid  by  the  above  method.  This  is  the  process 
which  has  been  adopted  in  the  latest  form  of  lunar  theory.  It  is  also  possible 
by  giving  m  other  values  to  trace  the  development  of  the  whole  family  of 
periodic  orbits  of  lunar  type.  These  orbits  are  of  great  theoretical  interest, 
especially  for  larger  values  of  m.  But  it  is  evident  that  the  effect  of  the 
neglected  parallactic  terms  will  become  more  considerable,  and  such  results 
may  differ  sensibly  from  true  solutions  of  the  restricted  problem  of  three 
bodies.  Also  when  m  exceeds  ^  the  question  of  convergence  begins  to  in- 
troduce practical  difficulties  and  the  method  of  quadratures,  followed  by 
Sir  G.  H.  Darwin  and  others,  becomes  necessary. 


229-231]  Lunar  Theory  I  265 

230.  To  find  the  value  of  a  recourse  must  be  had  to  an  equation  of 
motion  which  has  not  been  reduced  to  a  homogeneous  form  in  u,  s.  Since 
fl  =  z  =  0  and  r2  =  us,  the  first  of  (2)  becomes  in  the  present  case 

(D2  +  2mD  +  f  m2)  u  +  f  m"s  =  KU  (us)  ~ 
or 

a  2  {(2t  +  I)2  +  2m  (2t  +  1)  +  f  m2}  a^i+l  +  f  m2a  2  o^-21'-1  =  «w  (us)  ~  \ 


This  equation  must  hold  for  all  values  of  £,  including  f  =  l.    Then  w=s= 
and  therefore 

a  2  {(2i  +  1  +  m)2  +  2m2}  a2i  =  /ca~2  (S  a^)~\ 

But  (§  224)  *  =  /*  (w  -  rc')~2  =  ft  (1  +  m)2  rr\  so  that 

w2aa  =  fi  (1  +  m)2  (S  a*)-*  [S  !(2i  +  1  +  m)2+  2m2}  a*]-'  ......  (11) 

It  has  been  usual  to  write  w2as=  /A,  a  being  the  mean  distance  which  would 
correspond  to  the  mean  motion  n  in  the  absence  of  solar  or  other  perturba- 
tions. Thus  a  =  a  (1  +  powers  of  m)  when  the  values  of  a^  are  inserted. 
The  precise  form  of  this  relation  is  required  only  when  it  is  desired  to 
compare  two  theories  expressed  in  terms  of  a  and  a  respectively.  The  con- 
stant a  fixes  the  scale  of  the  orbit  and  therefore  depends  on  the  parallax, 
which  is  observed  directly. 

When  the  coefficients  a^,  and  a  have  been  determined,  (8)  gives  the 
value  of  C,  if  it  be  required. 

For  the  transformation  to  polar  coordinates, 

r  cos  (v  —  nt  —  e)  =  r  cos  (v  —  n't  —  e  —  £)  =  X  cos  £  +  F  sin  £=  \  (u%~1  +  s%) 
r  sin  (v  —  nt  -  e)  =  r  sin  (v  —  n't  —  e  —  £)  =  Y  cos  %  —  Xsmi-  =  %  (s£  —  u^~l)  i 
where  e  =  e'  —  (to  —  n')  t0>  since  £  =  (n  —  ri)  (t  —  t0)  and  ig  =  log  £     Hence 

r  cos  (v  —  nt  —  e)  =  a  {1  +  (a2  +  a_2)  cos  2f  +  (a4  +  a_4)  cos  4^  +  .  .  .H 

l**\          / 

r  sin  (v  —  nt  —  e)  =  a  {       («2  —  a_2)  sin  2£  +  («4  —  a_4)  sin  4£  +  ...  N 

which  lead  to  the  determination  of  r  and  v,  the  more  simply  because  v  —  nt  —  e 
is  evidently  of  the  second  order  in  m. 

231.  The  use  of  rectangular  coordinates  is  a  distinctive  feature  of  Hill's 
method.  But  for  some  purposes  polar  coordinates  present  advantages.  By 
a  simple  change  of  units  and  notation  (1)  become 


—  ?  +  2°^=  -^ 
d£2         d*         r3 

which  can  be  reduced  to  canonical  form  by  putting  (cf.  §  216) 


266  Lunar  Theory  I  [OH.  xx 

The  transformation  to  new  variables,  r,  I  ;  r',  I',  defined  by 
p  —  r  cos  I,     p'  =  r  cos  I  —  r~l  I'  sin  I 
q  =  r  sin  I,     q'  =  r'  sin  I  +  r"1  1'  cos  I 

will  leave  the  canonical  form  unchanged,  since 

p'dp  +  q'dq  -  (r'dr  +  I'dl)  =  0 

and  therefore  it  is  an  extended  point  transformation  (§  125).  Let  t  be 
eliminated  from  the  equations  by  taking  I  as  the  independent  variable. 
After  writing  out  the  equations  in  explicit  form  make  the  transformation 

r  =  l/cr,     r'  =  p/o;     T  =  w/a2 
and  finally  put  e  =  <r\     The  result  is  to  give  the  equations 

(—  Da—  ^ 

(o>  -  1)  ~  =  w2  -  p2  +  f  cos  2Z  +  i  -  e 
di 

(to  —  I)-™-  =  —  2pco  —  I  sin  21 
ctl 

and  the  integral  H  =  h  becomes 

£/>2  +  £  (&)  -  If  -  f  cos2  Z  -  (Aef  +  e)  =  0. 
Assume  a  solution  in  the  form 

p  =  i  2  a2ne2Ml/k,     w  =  2  b2ne2inl'k,     e  =  I  c2ne2Ml'k. 

—  00  —  00  —  00 

For  a  periodic  orbit  described  always  in  one  direction  as  regards  £  these 
series  are  convergent,  and  if  the  coefficients  are  real,  ayn  =  —  «_<*„,  b.2n  —  6_2n, 
Can  =  C-2H,  and  therefore 

_  1  dr  _         «  2nZ 

p~rdt~     ^fa^sll-y 

dJ  _  S  ,          2?iZ 

<o  =  1  +  -r.  —  b0  +  2  Z  62n  cos  -j- 
at  i  /? 


The  index  lc  is  arbitrary.  It  may  be  proved  that  if  k  is  an  odd  integer 
the  orbit  is  completed  in  k  circuits  and  is  symmetrical  about  both  axes,  and 
if  k  is  an  even  integer  the  orbit  is  completed  in  %k  circuits  and  is  sym- 
metrical about  the  axis  of^  only.  For  Hill's  variational  curve  k—l. 

The  substitution  of  the  assumed  series  in  the  equations  leads  to  three 
series  of  equations  which  must  be  solved  by  continued  approximation  as  in 


231,  232]  Lunar  Theory  I  267 

Hill's  method.  A  most  interesting  result  is  that  the  series  for  e  converges 
with  exceptional  rapidity,  so  that  the  equation 

r~3  =  c0+  2c2  cos  21 

where  c0  =  93c2  nearly,  represents  the  variational  curve  with  an  error  which 
on  the  scale  of  the  lunar  orbit  is  no  more  than  half  a  mile.  No  simpler  idea 
of  the  nature  of  this  curve  could  possibly  be  given. 

It  may  be  left  as  an  exercise  to  the  student  to  fill  in  the  details  of  the 
outline  conveyed  in  this  section*. 

232.  The  method  by  which  the  variational  curve  can  be  determined 
with  any  required  degree  of  accuracy  has  been  fully  explained.  But  it  must 
not  be  supposed  that  this  curve  represents  the  lunar  orbit  in  any  true  sense. 
It  is  merely  a  particular  solution  of  equations  which  are  themselves  only 
a  degenerate  form  of  those  which  characterize  the  Moon's  motion,  and  the 
only  significant  parameter  involved  is  the  mean  motion  of  the  Moon.  The 
next  step  is  to  seek  the  form  of  the  general  solution  of  the  same  equations. 
With  this  object  it  is  necessary  to  study  the  variation  of  the  particular 
solution  and  to  determine  a  fundamental  quantity  c. 

With  some  change  of  notation  (3)  and  (4)  of  §  214  give 

,J2 

~SN  +  ®SN=0  ...........................  (13) 


where,  in  the  application  to  (1), 

~ 


nj  - 


SN  being  the  normal  displacement  to  the  variational  curve,  -\Jr  the  inclination 
of  the  tangent  to  the  axis  of  X,  and  F  the  relative  velocity.  In  terms 
of  u,  s, 

V2  =  X*+Y2  =  us  =  - 


since  d/dt  =  ivD.     Hence,  R  being  the  radius  of  curvature, 


Also 


V  dV    ~  V  dt  \2  V   dt  J     dt  \2V*    dt  J  ^  4F4  V  dt  ) 
/DV-\       n  fDV™\ 

~      U^v      \tr»J 


u      Ds 

Cf.  J.  F.  Steffensen,  Eoyal  Danish  Academy,  Forhandlinger  (1909). 


268  Lunar  Theory  I  [OH.  xx 

Finally 


Therefore,  since  v  =  n'  —  n,  n'  =  mv  and  p  =  KV-, 

(    /J)za     L 

y-2®  =  -/c/ri-m2  +  2U(-Fl j 

(/  \  Du      La/         j 

-if^  +  ^Y (14) 


Now  since  u  =  ££0*  f*  s  =  £-*  So*  ?~2i  and  D  = 


and  t^  can  be  calculated  by  equating  coefficients  in 

S  (2i  +  I)2  o,i  ^+1  =  2  (2t  +  1)  a^'^1  .  S  f/i  ?«*. 

i  i  i 

Similarly,  by  the  first  of  (2)  when  H  =  0, 

u  (fcr~3  +  m2)  =  2u&Mi  ^  =  £2u  +  2m  Dw  +  ^m2  (5u  +  3 

i 
so  that 

it*  =  2  {(2t  +  I)2  +  2m  (2t  +  1)  +  |m2}  a2,  ^+1  +  f  m2 


whence  Mi  can  be  calculated  in  the  same  way,  When  Ui,  Mi  have  been 
found  it  remains  to  substitute  the  series  in  (14),  a  process  which  involves 
squaring  two  series,  and  the  result  may  be  written  in  the  form 


Thus  (13)  becomes 

£)Sy  ...........................  (15) 


and  the  derivation  of  ©t-  has  been  fully  explained.     It  is  easily  seen  that 
@_i  =  ®i  and  that  Mit  Ui  and  <B){  are  of.  the  order  j  2t    in  m. 

233.  Owing  to  the  symmetry  of  the  variational  curve  ©  is  a  periodic 
function  with  the  half  period  of  the  curve,  TT  /(•/?.  —  w').  Hence  by  §  215  one 
solution  of  (15)  has  the  form 

SAr=£°2^2'' 

and  c  is  the  quantity  which  is  now  required.     The  result  of  substituting 
this  series  is 

2  bj  (c  +  2j)2  ?+*  =  2  2  ®i  &,-_;  ?+* 

j  i  i 

which  must  be  an  identity,  and  therefore  for  every  value  of  j 

fy  (c  +  2jy 
or  more  fully,  since  ®,:  =  ©_<, 


232-234]  Lunar  Theory  I  269 

These  equations  are  of  infinite  order.  Nevertheless,  let  the  coefficients  b{  be 
eliminated  in  the  same  Avay  as  though  their  number  were  finite.  Then 
A  (c)  =  0  where  A  (c)  represents  the  determinant  of  infinite  order 


(c-  4)--<* 

>«           -  ©i 

-©2 

-  ©3                                -  @4 

4'-@0 

'           42  _  (S)         ' 

4 

!-©o' 

42-00             42-00       "• 

-©, 

(c-2)2-©0 

-©, 

-  ©2                 -  ©3 

22-©0 

2»-eo 

22 

-©o' 

22  -  ©o             2*  -  ©„ 

-e, 

-0, 

c2 

-©o 

-  0,                                 -   ©2 

02-©0 

02-©0 

O2 

-©o' 

02-©0                02-©0 

~©3 

-©2 

-©! 

(c  +  2)2-©0           -©, 

22-©0 

22-©0 

22 

-©o' 

22-©0  '   '       22-©0 

-©4 

-©3 

-©2 

-©!                      (C  +  4)2-©0 

42-©0 

42-©0 

42 

-©o' 

4"  -00              42-©0 

each  row  being  divided  by  such  a  factor  that  the  constituent  in  the  leading 
diagonal  becomes  1  when  c  =  0.  This  is  Hill's  celebrated  determinant, 
which  introduced  the  consideration  of  the  meaning  and  convergence*  of 
determinants  of  infinite  order  into  mathematical  analysis. 

234.  The  determinant  A  (—  c)  =  A(c),  for  the  change  only  reverses  the 
order  of  the  constituents  in  the  leading  diagonal.  Also  A  (c  +  2j)  =  A  (c), 
for  the  displacement  of  the  leading  diagonal  along  itself  may  be  compensated 
by  moving  the  divisors  of  the  rows.  Hence  if  c0  is.  a  root  of  A  (c),  +  c0  -f-  2j 
are  also  roots.  The  highest  power  of  c  in  the  development  is  given  by  the 
product  of  terms  in  the  leading  diagonal,  and  this  product  is 

A  M-  FT 

It  follows  that 


4J2-©0  (2}4-V®o)' 

=  (COS  7TC  —  COS  7T  V©o)/(l  —  COS  7T  V©o)- 
A  (c)  =  (COS  7TC  —  COS  7TC0)/(1  —  COS  7T  \/©o) 

for  this  contains  the  right  number  of  roots,  the  same  as  A0  (c),  and  the  same 
coefficient  of  the  highest  power  of  c.  The  roots  are  those  already  found,  and 
there  are  no  others.  But  this  equation  shows  that 

A  (0)  =  (1  -  COS  7TC0)/(1  -  COS  7T  \/©o) 

and  therefore  c0  is  a  root  of 

sin2^7rc0  =  A(0)sin2^7rV©o    .....................  (16) 

*  Cf.  Whittaker's  Modern  Analysis,  p.  35  ;  Whittaker  and  Watson,  p.  36. 


270  Lunar  Theory  I  [CH.  xx 

The  solution  of  A  (c)  =  0  is  thus  reduced  to  the  calculation  of  A  (0).  The 
latter  determinant  is  convergent  if  2;©*  is  convergent,  and  this  may  be 
assumed  for  sufficiently  small  values  of  rn. 

As  a  matter  of  fact  in  the  present  case  A  (0)  is  not  only  convergent  but 
very  rapidly  convergent.     It  may  be  written  in  the  form 

A  (0V- 


where 

Suppose  every  0,  to  be  multiplied  by  0i.  If  then  the  sign  of  0  be  changed 
the  sign  of  every  alternate  constituent  in  every  row  and  every  column  is 
changed.  Multiply  every  alternate  row  and  every  alternate  column  by  —  1 
and  the  original  determinant  is  restored.  This  involves  multiplication  of 
A  (0,  —  0)  by  an  even  power  of  —  1,  since  the  number  of  rows  and  columns  is 
equal.  Hence  A  (0,  —  0)  =  A  (0,  0),  and  A  (0,  0)  is  an  even  function  of  0. 
But  the  power  of  0  in  any  term  of  the  development  of  A  (0,  0)  is  the  sum  of 
the  suffixes  of  the  ©/  associated  with  it.  Therefore  the  sum  of  the  suffixes 
in  any  term  of  the  development  of  A  (0)  is  even.  Since  ©^  is  of  the  order 
j  2j  \  in  m,  this  means  that  the  order  of  every  term  is  a  multiple  of  4. 

It  is  evident  that  the  determinant  A  (0)  must  be  developed  axially,  the 
term  of  zero  order,  1,  coming  from  the  leading  diagonal  alone.  There  can 
be  no  term  in  ©,  alone,  for  ©,  incapacitates  by  its  row  and  column  two  units 
from  the  leading  diagonal  as  cofactors.  Similarly  a  product  ©;©,  incapaci- 
tates more  than  two  such  units  unless  their  rows  and  columns  intersect  on 
the  leading  diagonal.  Thus  i  =j  and  the  only  terms  of  binary  type  involve 
squares. 

235.  The  mode  of  developing  A  (0)  will  be  sufficiently  understood  if  m12 
be  neglected.  The  sum  of  the  suffixes  can  only  be  0,  2  or  4.  Hence  the 
only  possible  terms  are  of  the  type 

A  (0)  =  1  +  A®*  +  £022  +  C©!2©,  +  Z)©!4. 
It  is  also  easy  to  see  how  each  of  these  terms  arises.     Thus 


0 


0 


234,  235]  Lunar  Theory  I  271 

The  next  term  corresponds  to  three  consecutive  diagonal  constituents,  and 


c^e,  =  2 

i 


0 


0 


Finally,  the  term  in  ©x4  must  correspond  to  four  diagonal  constituents  only 
and  it  is  therefore 


0 


0 


0 


0 


D  =  22  &&•_!  /3//3/-1  =  A2 

i  j  J  3 

for,  as  the  two  minors  must  not  overlap,  i  cannot  have  the  values  J  or  j  ±  1. 

It  remains  to  calculate  the  values  of  these  coefficients.     Let   @0  =  4a2. 
Then 


*  32a  (2a  -  1)  Va  -  j     a  +j  -l)      J  32a  (2a  +  1)  (a  +j     a  -  j 
2,  1  1  1 


_  _  _=         _  _ 

_1  8a  (4a2  -  1  )  '  a  +  j     8a  (4a2  -  1)  (a 

TT  cot  ?ra          TT  cot  ^7rV@o 
80  4a2  -  1   = 


**    •      .„ 
x  a2  -  "2 


The  other  coefficients  can  be  calculated  similarly  by  first  reducing  to  the 
form  of  partial  fractions.  Hill's  results  include  all  terms  of  order  less  than 
16,  and  with  the  value  of  m  already  given  (§  229)  he  obtained  the  value 

c0  =107158  32774  16012. 

Without  going  further  than  the  term  of  which  the  form  has  actually  been 
found  here, 

............  (17) 


The  argument  given  above  as  to  the  order  of  the  terms  refers  to  ©x,  @2,  ... 
and  not  to  effects  arising  from  00.  But  1  —  60  is  itself  of  the  first  order, 
and  therefore  this  expression  neglects  m7  instead  of  m8.  Since  m  =  0'08  the 
error  in  c0  might  be  expected  to  occur  at  about  the  seventh  decimal  place, 
and  in  fact  it  is  about  5  units  in  this  place.  This  simple  expression,  involving 
only  @o  and  ®]  ,  is  therefore  very  approximate. 

It  may  be  noticed  that  +  ic  (n  —  n)  are  the  characteristic  exponents  of 
the  variational  curve.  Since  c  is  real  this  curve  represents  a  stable  orbit  for 
small  variations. 


272  Lunar  Theory  I  [OH.  xx 

236.  The  introduction  of  the  eliminant  of  infinite  order  was  a  bold  and 
original  expedient  on  the  part  of,  Hill,  though  justified  later  by  analysis. 
But  an  analogous  method  had  been  used  earlier  by  Adams,  whose  results 
were  published  after  the  appearance  of  Hill's.  They  refer  to  the  integration 
of  the  third  equation  of  (2)  when  H  =  0,  or 

D2z  -  z  (icr-3  +  m2)  =  0. 

If  z  be  neglected  in  the  coefficient  of  z,  that  is  in  r~:f,  the  series  already  used 
in  §  232  may  be  inserted,  and  the  equation  becomes 


which,  since  Mi  =  M_i  is  of  the  order  j  2i  \  in  m,  is  of  exactly  the  same  form 
as  (15).     A  solution  is  known  to  be  of  the  type 


and  g  must  be  determined  from  the  infinite  set 


Hence  the  eliminant  is  A'  (g)  =  0,  and  the  solution  is  given  by 

sin2  ^7Tg0  =  A'  (0)  sin2  \TT  \/(2^0) 
where  A'  (0)  is  the  result  of  replacing  ©,:  by  2Mt  in  A  (0). 

Adams  used  the  value  m  =  n'  /  n  =  0'0748013  exactly,  which  is  not  quite 
the  same  as  Hill's  value.     He  thus  obtained  the  corresponding  numbers 

m  =  0-08084  89030  51852,     g0  =  1-08517  13927  46869. 


CHAPTER    XXI 

LUNAR   THEORY   II 

237.  It  is  now  necessary  to  consider  the  form  of  the  general  solution  of 
the  equations  (6);  in  the  present  chapter  equations  will  receive  reference 
numbers  in  continuation  of  those  assigned  in  the  previous  chapter,  so  that 
the  latter  will  suffice  without  referring  specifically  to  the  chapter  or  section 
in  which  they  occur.  The  solution  of  (15)  may  now  be  written 

8^=^26^,     logft  =  t(n-  »')(*-  *0- 

The  arbitrary  constant  t:  makes  it  possible  to  assign  any  required  phase  to 
the  variation  in  relation  to  the  periodic  solution  and  as  8N  is  supposed  small 
(so  that  SN2  has  been  neglected)  the  coefficients  bi  may  be  considered  to 
have  a  small  arbitrary  factor.  These  two  arbitraries  make  the  small  variation 
otherwise  general.  Since  c  has  been  determined  it  would  clearly  be  possible 
to  determine  real  values  of  the  coefficients  (except  for  the  arbitrary  factor) 
by  substituting  the  series  in  (15),  equating  coefficients,  and  proceeding  by 
continued  approximation. 

Again,  if  So-  be  the  displacement  in  arc  corresponding  to  8N,  by  (2)  of 
§  214  adapted  to  the  present  notation, 


or  (§  232) 


B  irrt 

-yr-  -  -r-  +  2  m  )8N  =  -  1  VD   - 
\  Du      Ds          ] 


Hence,  V  being  an  even  function  of  £,  i8a  has  the  same  form  as  8N.     But 
since 


VeL*  =  ivDu,     Ve  ~  '*  =  iv  Ds 
and 

8N=  SX  sin  i/r  -  SF  cos  ^  =  fa  (8u  .«-*-&.  e1*) 

So-  =  SX  cos^  +  8  Y  sin  ^  =  %  (8lt  •  e~^  +  gs  •  e^ 
it  follows  that 

Su  =  ^  (8N  +  i8<j  ),     8s  -  V^S  (fa  -  SN). 

P.  D.  A.  18 


274  Lunar  Theory  II  [CH.  xxi 

Hence  8u,  &s,  like  Du,  Ds,  are  odd  functions  in  £  with  real  coefficients,  and  it 
is  possible  to  write 


the  coefficients  as  expressed  being  the  same  in  the  two  series  since  8u  4-  8s  =  28X 
is  real.  For  the  purpose  of  this  argument  it  is  necessary  to  associate  the  +  c 
solution  for  Bu  with  the  —  c  solution  for  Ss,  and  to  notice  that  (£j/Dic  are 
constant  conjugate  imaginaries  with  absolute  value  1  which  have  been  re- 
garded as  external  factors  of  the  series  with  real  coefficients  for  8N,  i&o-,  8u 
and  Ss.  At  the  same  time  8u  —  Ss  is  a  pure  imaginary. 

Hence  the  general  solution  of  (6),  differing  but  little  from  the  variational 
curve,  may  be  written 


i   p  i  p 

where  i  has  all  integral  values  between  +  oo  and  p  has  the  values  0  and  +  1. 
Also  A2i  =  a^  as  in  the  variational  curve  and  c  is  a  determined  function  of  m 
which  has  been  denoted  by  c0. 

238.  But  the  solution  which  is  now  sought  differs  by  a  finite  amount 
from  the  variational  curve.  The  above  form  must  therefore  be  regarded 
merely  as  the  beginning  of  the  full  development.  Hence  the  restriction  on 
p  will  now  be  withdrawn  and  its  values  will  be  allowed  to  range  between 
+  oo  .  The  coefficients  of  the  first  order  A2i±c  contain  a  small  arbitrary  para- 
meter e  and  the  higher  coefficients  A^pc  will  be  obtained  by  successive 
approximation  in  the  ordinary  way,  so  that  A2i±pc  will  be  of  the  order  p 
in  e.  The  introduction  of  e  into  the  solution  will  affect  both  A2t  and  c,  and 
&21  and  c0  represent  those  parts  only  which  are  functions  of  m  alone  and  not 
of  e. 

It  is  assumed  that  this  process  will  produce  convergent  series.  If  they 
converge  they  are  true  solutions  of  the  differential  equations,  and  not  other- 
wise. This  recurrent  question  in  dynamical  astronomy  cannot  be  dealt  with 
here.  But  the  reader  must  realize  its  fundamental  importance,  and  he  will 
understand  why  so  much  attention  has  been  given,  by  Poincare  especially,  to 
discussions  of  this  kind,  although  they  may  seem  unproductive  of  new  and 
striking  results. 

It  is  now  to  be  noticed  that 


^c)  =  (2i  +  1  +  pc)  £2i+l  £f°, 

and  therefore  that  the  result  of  putting  £i  =  £  will  affect  in  no  way  the  pro- 
cess of  calculating  the  coefficients.  If  this  substitution  is  made  it  is  only 
necessary  to  retain  c  explicitly  in  the  index  of  £"  and  to  remember  that  the 
argument  of  the  trigonometrical  term  corresponding  to  £2*+1+.Pc  is 

(2i  +  !)(»-  n')  (t  -  t0}  +pc  (n  -  ri)  (t  -  t,). 


237-239]  Lunar  Theory  II  275 

With  this  understanding  the  form  of  solution  becomes 


i-p0?i+pc  .........  (18) 

i   p  i  p 

Comparison  of  these  series  with  (7)  shows  immediately  that  the  effect  of 
substituting  in  the  differential  equations  and  equating  coefficients  of  £^'+2c 
will  follow  as  before  if 

A,     22,     2i+pc,     2j  +  qc 

i  p 

be  substituted  respectively  for 

a,        2,          2i,  2j. 

i 

Thus  to  (10)  corresponds  the  equation 

2  2  A2i+pc  pj  +  qc,  2i  +pc]  A^j+2i_qc+pc 

i    p 

+  [2j  +  ^c,  +]  A2j_2i_2+qc_pc  +  [2j  +  qc,  -]  A_2j^_z_qc_pc}  =0    ..  .(19) 

which  holds  unless  j  =  q  =  0.  The  form  of  the  symbolical  coefficients  has 
been  given  with  (10),  [2j  +  qc,  2j  +  <?c]  =  -l  is  the  coefficient  of  A0A2j+qc, 
and  [2j  +  qc,  0]  =  0  is  the  coefficient  of  A0A_2j_qc.  The  counterpart  of  (8)  is 

a-2(7=  22  {(2i  +  1  +pcf  +  4m  (2i  +  1  +pc)  -f  |m2}  A*#+po 


i  p 

•"• 


—  2i—  2—  pc  • 
i   p 


239.  Of  the  first  importance  are  the  terms  which  depend  on  the  first 
power  of  the  parameter  e.  When  8N2  was  neglected  A.2i  was  identical  with 
oy,  and  therefore  A.^  =  a.2i  when  e2  is  neglected.  Let 

AM+C  =  e  ei}     J.2i_c  =  e  e/. 

The  limitation  to  the  first  order  in  e  means  a  return  to  the  equations  at  the 
end  of  §  237  and  the  only  admissible  values  of  q  are  +  1.  With  either  value 
p  must  be  chosen  so  that  c  occurs  only  once  in  the  suffixes  of  any  term,  or 
terms  involving  e2  will  be  introduced.  Hence  (19)  gives 

2  pj  +  c,  2i  +  c]  a^2j+zi€i  4-  [2j  +  c,  2i]  a2t-e'_j+£ 

i 

Permissible  changes  in  i  make  it  possible  to  reduce  all  the  suffixes  of  e,  e  to 
the  form  i,  and  the  simpler  equations 

2  pj  +  c,  2t  -f  c]  a_2j+.,;€i  +  [2j  +  c,  2i  +  2j]  a2j+2j  e/ 

+  2  [2j  +  c,  +]  a^^ei  +  2  [2j  +  c,  -]  a_2,-_2;_2  e/}  =  0 

•"*  \\_"J        ^>  ^*        CJ  Qi—%j+y&fi   T  [- J        C,  Li  -f-  ZjJ  tt2£-(-2/fi 
i 

_j_  2  [27 c  +1  a.  •    •  .,€•'  4-  2  r2i  —  c  1  <x        •    6-1  =  0 

18—2 


276  Lunar  Theory  II  [CH.  xxi 

are  thus  obtained.  Since  the  numerical  value  of  m  is  introduced  from  the 
outset  and  c  has  been  determined,  the  coefficients  of  ei}  e/  are  numbers,  which 
in  general  become  rapidly  smaller  at  a  distance  from  the  central  term.  The 
equations  can  therefore  be  solved  by  continued  approximation.  As  they 
determine  the  ratios  only  of  e^,  e/,  it  is  possible  to  put 


i  e0,     €i  =   i  e0  +       e0. 

The  equations  for^'  =  +  1,  +  2,  ...  will  then  serve  to  determine  the  coefficients 
bi}  fr,  bi',  &',  where  b0  =  &  =  1,  &  =  b0'  =  0.  For  j  =  0, 

0=...+  [c,  2  +  c]a2ej     +  [c,  2]  a2e/      +  2[c,+]a_4e1  +  2[c,-]a_4e1'       ^ 

-«0e0  +  2[c,+]a_2e0+2[c,-]a_2e0'       1(21) 

-f  [c,  -  2  +  c]  a_2e_!  +  [c,  -  2]  a_2  e'_!  +  2  [c,  +]  a»  e_!  +  2  [c,  -]  a0  e'_j  +  .  .  .  J 

with  a  similar  equation  obtained  by  changing  the  sign  of  c  and  interchanging 
e.  e'.  Either  of  these  two  equations,  with  e0  —  e,'  =  1,  determines  e0  and  e0', 
and  hence  e{,  e/  in  general.  The  two  must  lead  to  the  same  result,  and 
together  are  merely  a  check  on  the  value  of  c,  which,  had  it  not  been  deter- 
mined otherwise,  could  in  theory  be  deduced  from  the  whole  set  of  these 
equations. 

240.  Before  continuing  the  development  of  a  method  the  whole  aim  of 
which  is  a  systematic  advance  towards  great  accuracy  in  the  complete  results, 
and  which  is  therefore  apt  to  obscure  the  main  features  of  the  actual  motion 
of  the  Moon,  it  will  be  well  to  consider  the  kind  of  results  which  have  already 
been  obtained  implicitly  or  can  be  readily  deduced.  For  this  purpose  a  low 
order  of  approximation  must  be  adopted  and  m4  will  be  neglected.  Then  it 
is  easily  found  that 

o,  =  [2,  +]  =  TVn2  +  im3,     a_2  =  [-2,  -]=-!£  m2-fm3 
21f0  =  1  +  2  m  +  f  m2,     2  J/,  =  2M,,  =  f  m2  +  -L9-m3 
U0  =  1,     U,  =  fin2  +  3m3,     U^  =  -  J£ms  -  -^m8 
+  2  (U0  +  m)2  =  1  +  2m  -  -|m2 


To  the  order  named,  the  combination  of  (16)  with  (17)  gives 

c0  =  v©o  +  ie12/(i-@0)v©o 

=  1  +  m  -  f  m2  -  -^-m3  =  1  "07263 
and  similarly 

go  =  V(2if0)  +  M?l(l  -  2M0)  vWo) 

=  1  +  m  +  f  m8  -  ffm3  =  1-08521. 

The  numerical  value  of  g0,  corresponding  to  m  =  0*08085,  is  much  nearer  the 
truth  than  that  of  c0.     Also  it  follows  from  (11)  that 


239-241]  Lunar  Theory  II  277 

Then  (12)  give 

r  cos  (v  —  nt  —  e)  =  a  (1  —  (m2  +  f  m3)  cos  2£} 

r  sin  (v  —  w£  —  e)  =  a  (V1112  +  -<f  m3)  sin  2£ 
whence 

'  v  =  nt  +  e  +  (V-™2  +  -if  ra3)  sm  2f 


r  =  a  (1  -  i  m2  +  £  m3  -  (m2  +  £m3)  cos  2£}. 

Terms  depending  on  m  only  are  called  variational  terms.  The  coefficient  of 
the  principal  term  of  the  variation  in  longitude  is  thus 

JgLm2  +  -V3-m3  =  0-01013  =  2090" 

which  is  some  16"  in  defect  of  the  true  value.  This  term  was  discovered 
observationally  by  Tycho  Brahe,  and  its  period,  indicated  by  2£  (or  2D  in 
Delaunay's  notation),  is  half  a  synodic  month. 

241.     The  equations  (20)  for  j  =  ±  1,  when  the  leading  terms  only  are 
retained,  become  simply 

e,  =  {[2  +  c,  c]  a_2  +  2  [2  +  c,  +]}  e0  +  [2  +  c,  2]  a2  e0' 
e_!  =  [-  2  +  c,  c]  a2  e0  +  {[-  2  +  c,  -  2]  a_2  +  2  [-  2  +  c,  -]}  e0' 
e/  =  [2  -  c,  2]  a,  e,,  +  {[2  -  c,  -  c]  a_2  +  2  [2  -  c,  +]  }  e0' 
e'_1={[-2-c,  -2]a_2+2[-2-c,  -]}e0  +  [-2-c,  -c]a,e0'. 

It  is  to  be  noticed  that  [x,  y\,  [x,  ±]  contain  as  a  divisor 

Dx  =  2x2  -  2  -  4m  +  m2 

and  that  this  has  the  factor  m  when  +  x  —  2  —  c.     It  is  easily  found  that 
[2  +  c,c]  =  -&,     [2  +  c,  2]=  -I,     [2  +  c,+]  =  T^m2 


[-  2  +  c,  -]  =  |f  m  +  ffgm',     [2  -  c,  +]  =  -  if  m  -  ffim2 
[2-c,2]  =  -im--ff,     [2-c,-c]=-irn--/ff 

as  far  as  the  present  low  order  of  approximation  requires.     Hence  with  the 
approximate  values  of  a2,  a_2, 


It  has  been  seen  how  the   order  of  e_15  ex'  is  lowered  by  the  divisor  Dx 
A  similar  circumstance  affects  the  coefficients  of  (21)  more  seriously,  since 

Dc  =  2c2  -  2  -  4m  +  m2  =  - 


278  Lunar  Theory  II  [CH.  xxi 

The  disappearance  of  the  terms  below  m3  explains  why  an  extremely  accurate 
value  of  c  is  required  in  the  numerical  development.  Without  continuing 
the  series  for  c  beyond  m3,  Dc  is  here  limited  to  a  single  term,  and  therefore 
only  the  terms  of  the  very  lowest  order  in  (21)  can  be  taken  into  account. 
This  equation  is  thus  reduced  to 

[c,  2]  a.2e/  -  e0  +  [c,  -  2  +  c]  rt_2e_,  +  2  [c,  +]  «„€_,  =  0 
where 

[c,  2]  =  [c,  -  2  +  c]  =  -  ^m-3,     [c,  +]  =  -  f^m-1. 
Hence 

A  (A  e«  +  &  60')  -  60  +  (j&  -  T%)  (&  e0  +  -w-  O  =  o 

which  gives  quite  simply  3e0  +  e0'  =  0,  and  with  e0  —  e0'  =  1,  e0  =  ],  e0'  =  —  f  . 
These  values,  though  representing  only  the  terms  of  zero  order  in  m,  are  true 
within  1  per  cent.  It  follows  that 


e/^ifm  +  ffim2,     eL^-^m' 

where,  owing  to  the  imperfect  values  of  en,  e,,',  the  second  terms  in  e_!,  e/  may 
also  be  defective. 

242.     The  terms  thus  found  in  (18)  are 

u  =  ae£(eb£°  +  ^^  +  e^  +  <>-£-*+*  +  e/^  +  eL,^) 
s  =  ae^1  (e£~c  +  *&  +  e£-z~c  +  ^2~c  +  ^'^+c  +  e'-i^0) 
to  which  correspond  (§  230) 
rcos(v  -  nt  -  e)  =  ae  {(60  +  e0')cos 
rsin(w  -  nt-  e)  =  ae{(e0-  e0')sin 

where 

<f)  =  c(n  —  n')(t  —  ti) 

is  the  argument  of  the  trigonometrical  term  corresponding  to  £c.     These 
terms  are  additive  to  the  variational  terms  already  obtained. 

The  fundamental  terms  are 

r  cos  (v  —  nt  —  e)  =  a  (1  —  |e  cos  <f>) 
r  sin  (v  -  nt  —  e)  =  ae  sin  0. 

Now  in  elliptic  motion  (24)  and  (25)  of  Chapter  IV  give,  to  the  first  order 

in  e, 

r  cos  iv  =  a  (—  f  e  +  cos  M  +  ^e  cos  2M  ) 

r  sin  w  =  a  (  sin  M  +  |e  sin  2M) 

r  cos  (w  —  M)  =  a  (1  —  e  cos  Jlf  ) 
r  sin  (w  —  M}  —  2ae  sin  -M. 


241-243]  Lunar  Theory  II  279 

These  can  be  identified  with  the  former  by  putting  a  =  a,  e  =  2e,  <j)  =  M,  and 

v  =  nt  +  e  -f  w  —  M 

—  w  +  {n -  c (n  —  n'}}  t  +  e  +c(n  —  n') ti 
=  w  +  {1  -  c/(l  +  m)}  nt  +  const. 

This  shows  that  to  this  extent  the  motion  of  the  Moon  is  purely  elliptic,  with 
eccentricity  |  e,  but  that  this  motion  is  referred  to  a  line  rotating  uniformly, 

given  by 

y0  =  {l-c/(l+m)}wi  =  (fm2  +  ^m3+...)^. 

Thus  c  determines  the  motion  of  the  lunar  perigee,  which  completes  a  revolu- 
tion in  the  direct  sense  in  rather  less  than  9  years.  The  above  approximation 
gives  128  sidereal  months  or  3500  days. 

In  the  older  lunar  theories,  beginning  with  Clairaut,  the  rotating  elliptic 
orbit  is  adopted  in  the  first  approximation. 

243.     The  result  of  collecting  the  terms  found  so  far  as  necessary  is 
r  cos  (v  —  nt  —  e)  =  a  [1  —  m2  cos  2£  —  ie  cos  <j) 

-  (if  m  +  -W™2) e  cos  (2£  -  4>)  +  /im2e  cos 
r  sin  (v  —  nt  —  e)  =  a  {^m2  sin  2£  +  e  sin  <f> 

+  (if  m  +  J^m2)  e  sin  (2£  -  <£)  +  /ff  mae  sin 

The  effect  of  dividing  the  latter  by  the  former  is  to  add  to  the  second  series 
the  terms 

m'2e  (cos  2£  sin  <f>  +  {±  sin  2£  cos  <£)  =  m2e  {f|-  sin  (2£  +  <£)  -  £  sin  (2£  -  <£)}. 
Hence  the  longitude  is  approximately 
v  =  nt  +  e  +  -^m2  sin  2£  +  e  sin  <f> 

+  (JgB-m  +  Wm2)  e  sin  (2£  -  <£)  +  iim2e  sin  (2^  +  </>)• 

As  a  constant  of  integration  introduced  at  one  stage  of  the  present 
method,  e  may  be  defined  in  any  suitable  way  for  the  later  stages.  Its 
value  depends  on  the  exact  definition  adopted  and  will  be  found  by  com- 
paring the  final  results  with  observation.  Thus  |e  as  defined  by  Brown  is 
not  to  be  identified  with  the  e  of  Delaunay,  for  example.  The  difference  is 
not  great,  however,  and  its  value  may  be  taken  to  be  0'0549!  Thus  the  co- 
efficient of  the  principal  elliptic  term  in  longitude,  e  sin  <£,  is  of  the  order  6°'3. 

The  term  next  in  importance  has  the  argument  2£  —  <f>  (or  2D  —  I  in 
Delaunay 's  notation).     The  coefficient  is  right  to  the  order  given,  though  the 
above  derivation  left  this  doubtful,  and  its  value  gives 
(j^m  +  ^-m2)  e  =  73'  nearly. 

The  true  coefficient,  depending  on  e  alone,  is  4608".  This  inequality  is 
the  largest  true  perturbation  in  the  Moon's  motion  and  is  known  as  the 
Evection.  Its  discovery  from  observation  is  due  to  Ptolemy. 


280  Lunar  Theory  II  '  [CH.  xxi 

The  term  with  the  argument  2f  -f  (f)  (or  2Z)  +  I)  is  much  smaller.  The 
above  coefficient  gives  157",  while  the  true  value  is  about  175"  for  the  part 
depending  on  e  alone.  It  will  be  noticed  that  the  greater  part  of  it  is 
due  not  to  a  true  perturbation  in  the  rectangular  coordinates  but  to  inter- 
ference between  the  variation  and  the  principal  elliptic  term  in  deriving  the 
longitude. 

244.  The  terms  depending  on  the  first  power  of  the  solar  eccentricity  e' 
will  be  next  considered.  With  z  =  0  and  the  solar  parallax  still  neglected, 
H  =  f}2  and  (4),  (5)  become 

D2  (us)  -  Du  .Ds  +  2m  (sDu  -  uDs)  +  f  m2  (u  +  s)*=C-  3H2  +  D~l  (Dt  O,) 


D  (uDs  -  sDu  -  2mws)  +  f  m2  (u2  -  s2)  =  s        -  u 

9s  du 

where  (3)  gives 

H2  =  m2  ^  (3r*S*  -  r2)  -  £m2  {3  (u  +  s)2 


Now 

rS  =  (XX'  +  YY')  rr1  =  i  (u  +  s)  cos  %  -  1  1  (u  -  s)  sin  %' 

where  (§  223)  %  =  v'  —  n't  —  e'  =  v'  —  <f>'  is  the  solar  equation  of  the  centre. 
Hence 

i*S*  =  I  (u2  +  s2)  cos  2%  +  iws  -  i*  (u2  -  s-)  sin  2%' 
and  therefore 

a'3 

I12  =  m2  —  {f  (u-  +  s2)  cos  2%  +  -«us  -  ft  (u-  -  s*)  sin  2%'}  -  |m2  (3u-  +  3s2  +  2««) 
*"i 

where  u,  s  have  the  values  given  by  the  variational  curve.  The  Sun's  mean 
anomaly  is 

$  =  ri(t-  ts)  =  m(n-  M')  (t-ta)  =  -i  log  ^3m. 

The  whole  disturbing  function  must  ultimately  be  developed  in  powers  of 
f3m  as  far  as  necessary,  the  coefficients  involving  u,  s,  a'"1  and  e'.  But  for  the 
immediate  purpose  it  is  easily  verified  that  to  the  first  order  in  e', 

£v  W  C\         t  1  C\      I  I/  Cv  *'-\/  A        /  if 

—  =  —  :  cos  2y  =  1  +  3e  cos  d>  ,      —  sm  2v  =  4e  sin  c6  . 

M  5  ««  •  /V  '/y»o  /*/ 

Hence 

n2  =  fmv  {u2  (-  Ksm  +  K3-m)  +  *2  (K3m  -  i&-m)  +  us  (£3m  +  r3-m)} 

A«2  -  f  mV  {u2  (-  ^3m  -  f  &-»)  +  s2  (ft3m  +  if,—)  +  us  (^  -  r3-m)]- 

Thus  the  right-hand  members  of  the  equations  at  the  beginning  of  this 
section  will  be  of  the  form 


a         *m  2m,    a 


for,  as  in  §  238,  the  suffix  of  £3  may  be  suppressed  in  the  calculation  with  the 
proper  understanding  as  to  the  argument  corresponding  to  £m  in  the  results. 
The  solution  is  of  the  form 


u  =  a  ,ipm*,    8  =  a- 

i    p 


243-245]  Lunar  Theory  II  281 

where 

A2f  =  a.,,,     A2i+m  -  e'rj,:  ,     A  2,-_m  =  e'rj' 

and  p  has  the  values  0,  +  1  only,  until  higher  powers  of  e  are  taken  into 
account.  The  solution  follows  the  same  course  as  in  §  239  except  that  there 
are  now  terms  on  the  right-hand  side  of  the  equations.  The  equations  of 
condition  corresponding  to  (20)  are  thus 

2  {[2j  +  m,  2i  +  m]  a_2j+2;7?;  +  [2j  +  m,  2i  +  2j]  o^V 
i 

+  2  [2/  +  m,  +]  a2/-_2;_2  ^  +  2  [2j  +  m,  -]  a_2j_2(;_2  T//}  =  E\j+m. 

This  form  results  from  the  linear  combination  of  a  pair  of  equations  obtained 
by  comparing  coefficients  of  £2J+m  and  in  these  the  leading  terms  by  analogy 
with  (9)  are  respectively 

.  .  .  +  {4/2  +  2/  +  1  +  4m  (/  +  1)  +  f  m2]  a0e'Vj 

+  {4/2  -  2/  +  1  -  4m  (/  -  1)  +  f  m2}  au«V-;  +  •  •  •  =  *'&*+* 
...  -  4/  (1  +/  +  m]  a0e'rij  -  4/  (1  -/  +  m)  ao^Y-;  +  •  •  •  =  e'E'2j+m 

where  /  is  written  for  j  +  -|  m.  The  combination  is  such  that  the  coefficient 
of  ?;'_;  vanishes  and  that  of  rjj  becomes  —  1.  Hence 

rv,        =  ¥  (1  -/  +  m)  K2j+m  +  {4,f*  -  2j'  +  I  -  4m  (f  -  1)  +  f  m2}  E'2j+m 

¥2(8/2-2-4m  +  m2) 

The  divisor,  which  appears  also  in  the  symbolical  coefficients  [  ],  becomes 
small  only  through  the  factor/,  whenj  =  0,  4/a  =  m2. 

245.  The  calculation  of  77,-,  ?//  when  m  is  given  its  numerical  value  at 
the  outset,  proceeds  as  in  the  case  of  e,-,  e/  with  this  difference,  that  the 
equations  contain  definite  right-hand  members.  A  particular  solution  of  the 
differential  equations  is  required,  representing  a  forced  disturbance  of  the 
steady  variational  motion.  Hence  no  new  constant  of  integration  enters. 

The  machinery  is  of  course  absurdly  elaborate  when  only  the  main  parts 
of  the  leading  terms  are  sought,  but  this  plan  will  be  pursued.  It  is  easily 
found  that 

O2  =  f  mVa2  {-  \  (^+ni  +  £-2-m)  +  \  (£2~m  +  £-2+tt'  +  (1  +  6a_2      m  +    -m 


with  the  neglect  of  m  in  the  coefficients  of  £±2±m,  but  not  £±m.  The  operator 
Dt  applies  to  £"±m  only  and  gives  a  multiplier  +  m  to  every  term,  while  the 
operator  D~l  applies  to  £  generally  and  gives  divisors  ±  2  +  m  or  +  m.  Hence 
to  the  same  order  in  m 


Also 

an2      an, 

s  -r w  — —  =  £m2e  £ 


282  Lunar  Theory  II  [CH.  xxi 

Hence 

#m  =  E-m  =  -  f  m2  (1  +  6a_.2),     Emf  =  -  £"_m  =  12m2a_2 

Em"  =  (-  m-1  +  f)  Em  -  ^m--Em'  =  f  m  +  ^m2 
E"_m  =  (m-1  -  I)  E_m  -  im-2#'_m  =  -  f  m  -  Sfm*. 

Thus  ??0,  770'  must  be  of  the  first  order  in  m  and  give  rise  to  terms  of  at  least 
the  third  order  in  the  equations  for  j  =  ±  1.  These  contain  no  small  divisor 
and  for  the  lowest  order  they  give  immediately  : 

—  Vi     =  E"2+m    =  |  -fi"2+m  =  ^  m2 

—  rji    =  E"2_W    =  |£"2_m  =  —  Mm2 


Coefficients  of  the  form  [m,  y]  are  of  the  order  —  1  in  m,  but  they  multiply 
terms  of  at  least  the  fourth  order  in  the  equations  for  j  =  0.  These  give 
therefore  to  the  second  order 

-770      +  2  [m,  +]  a.rj^      +  2  [m,  -]  a^'^  =  E"m 

-i)0'  +  2  [-  m,  +]  OoV-i  +  2  [-  m,  -]  a077_,  =  E"_m 
where 

[m,  +]  =  [-  m,  +]  =  -  f  ,     [m,  -]  =  [-  m,  -]  =  f  . 
Accordingly 

-7;0  =  fm-fm2,     -770'  =  -fm  +  fm2. 

Thus  the  principal  terms  depending  on  the  solar  eccentricity  may  be  put 
in  the  form 

r  cos  (v  —  nt  —  e) 

=  ae'  {(770  +  T70')  cos  <£'  +  (77,  +  V-O  cos  (2£  -f  <j>')  +  (^  +  77^)  cos  (2£  - 
=  ae'  (fm2  cos  <£'  +  ^m2  cos  (2^  +  <£')  -  |m2  cos  (2£  -  </>')} 

r  sin  (v  —  nt  —  e) 

=  ae'  {(770  -  V)  sin  f  +  (^  -  77/_])  sin  (2|  +  f  )  +  (j?/  -  77^)  sin  (2f  - 
=  ae'  {-  3  (m  -  m2)  sin  <£'  -  |im2  sin  (2f  +  <£')  +  ffm2  sin  (2f 


In  deriving  the  longitude  there  are  no  interfering  terms  of  this  order,  and 
the  last  line  without  a  gives  the  additional  terms  depending  on  e'.  The 
term  with  argument  0'  (or  Z7)  is  called  the  Annual  Equation  after  its  period. 
The  value  of  e  is  0'01675  and  the  coefficient  of  this  part  of  the  term, 

—  3e  (m  —  m2),  is  —  770"  as  compared  with  the  complete  value  —  659".     For 
the  argument  2£-<£'  (or  W  -  I')  the  coefficient  ffe'm2  is  +  109",  the  true 
value  being  +  152",  and  for  the  argument  2£+  0'  (or  2D  +  I')  the  coefficient 

—  -j^e'rn2  is  —  15"'5,  the  true  value  being  —  21"'6.     The  discrepancies  are 
considerable  and  show  that  the  parts  depending  on  higher  powers  of  m  are 
large.     As  series  in  m  the  coefficients  converge  slowly,  and  hence  the  great 


245,  246]  Lunar  Theory  II  283 

advantage  of  the  Hill-Brown  method,  which  by  employing  an  accurate 
numerical  value  of  m  from  the  beginning  avoids  expansions  in  this  parameter 
altogether. 

246.     In  deriving  the  terms  with  the  characteristic  a'"1  alone,  e  is  neg- 
lected and  therefore  fla  =  0,  Dt£l  =  0,  and 

H  =  a,  =  2m2  a'-1  Par3  =  naV"1  (or'S3  -  Sr'jS)  . 
=  ^m'a'-1  [5  (M  +  s)3  -  llus  (u  +  s)} 

since  rS  =  X  =  £  («  +  ,9)  when  e'  =  0.  The  terms  on  the  right-hand  side  of 
(4),  (5)  are  thus 

-  4O:J  =  -  ^m'a'-1  {5  (u3  +  s3)  +  3ws  (u  +  s)}  =  a?  a'-1 


-  u      3  =  -  fmV-1  (5 


-  s3)  +  us  (u  -  s)}  =  a3  a'-1 


respectively.     The  additional  terms  required  in  the  solution  must  be  of  the 

form 

n,  _  os^'-i  r?n       j-2i+i      (,  _  a2<7/~1  /  —  J  T»    •     J*2*+i 
M  —  a  a      c,  ^a2i+1  t,      ,     *  —  a,  a,      c,     z(«_2t_1  1, 

in  order  to  produce  odd  powers  of  £  Similarly  H4  has  the  factor  a'~2  and 
gives  rise  to  terms  with  the  same  arguments  as  the  variational  terms.  The 
solution  follows  the  same  course  as  for  the  terms  with  characteristic  e*,  and 
the  relation  connecting  E"2j+1  with  E2j+1,  E'2j+l  is  the  same  as  before  when 

y=j+i 

The  principal  terms  are  given  by  2j  +  1  =  +  I,  ±  3.  The  divisor  D.2j>  is  of 
the  order  m  when  j'  =  ±  |  only.  But  fls  contains  m2  as  .a  factor.  Hence, 
when  terms  of  the  order  rn3  are  neglected  in  E'2j+1,  m2  can  be  neglected 
in  m~2ns  and  the  variational  coefficients  a2,  a_2  are  not  required.  Thus  it  is 
enough  to  write 

-  403  =  -  $  m'aV-1  {5  (£»  +  ^ 


and  therefore 


—  tt_s =  E  f—9  =  — ^r  E—?,  H — 1—  E'_ .j  =      -4p-  m2. 
Also,  to  the  same  order  in  m, 

E"     =  (Zm-1  4-ii"l  E     4-  (—  4m"1  —  -2-7-^  E'  ,  =  — 44m— ^4§m2 

j->     — j  —  i  ^  in         \^  if/  *^^ — 1    *^  V        4  f  fi  /  "^^  — 1  ^^         ^^  1  ^  fi         * 

The  equations  for  a1(  a_j  can  be  adapted  from  (21)  and  its  correlative  by 
putting  c  =  l,  e0  =  €/  =  «!.  and  e0' =  €_!  =  «_!.  To  the  second  order  in  m 
these  give 

[1,  2]  a,^  -a,  +  [I,  -  1]  a_2  o_,  +  2  [1,  +]  a0a^  =  E," 

[-  1,  1]  a.^  -  a_x  +  [-!,-  2]  a_2a_,  +  2  [-  1,  -]  a,*.,  =  ^"^ 


284  Lunar  Theory  II  [en.  xxi 

whence 


^  =      in  +  i     m2 


and  therefore 

-  «i  =  Mm  +  ftm2>     -a-i  =  -  ft  m  -  -W-  m2- 
The  additional  terms  in  their  elementary  form  are  thus 

r  cos  (v  -  nt  —  e)  =  a2a'~3  {(«i  +  a_x)  cos  £  +  (a3  +  a_3)  cos  3f} 
=  a2  a'-1  {(|f  m  +  ^m2)  cos  £  -  ff  m2  cos  3£j 
r  sin  (v  —  nt  —  e)  =  a2a/-1  {(ctj  —  «_,)  sin  £  +  («3  —  a_3)  sin  3£} 

=  a2  a'-1  {  -  (->£  m  +  ^  m2)  sin  £  +  if  m2  sin  3£} 

and  the  last  line,  divided  by  a,  gives  the  corresponding  terms  in  longitude. 
The  mean  parallax  of  the  Sun  is  8"*80  and  of  the  Moon  3422//<7  ;  to  the 
above  order  a/a'=  0*002571.  This  gives  —114"  for  the  coefficient  of  the 
first  term  (argument  £  or  D}  and  l//-6  for  the  coefficient  of  the  second 
(argument  3£  or  3D),  whereas  the  complete  values,  with  the  characteristic 
a/a'  alone,  are  —  125"  and  under  1".  The  term  with  argument  D  is  known 
as  the  Parallactic  Inequality.  Its  period  is  one  lunation  (or  synodic  month) 
and  the  comparison  of  its  theoretical  coefficient  with  observation  gave 
probably  the  best  determination  of  the  solar  parallax  until  the  direct  geo- 
metrical method  based  on  the  observation  of  minor  planets  was  adopted. 
This  use  of  the  parallactic  inequality  is  not  entirely  free  from  objection 
because  the  Moori  cannot  be  observed  throughout  a  complete  lunation  and 
systematic  error  may  be  suspected,  due  to  the  varying  illumination  of  the 
lunar  disc. 

247.  Hitherto  the  terms  of  u,  s  which  are  of  the  first  order  in  the 
characteristics  e,  e',aa'-1  have  alone  been  considered.  If  the  third  coordinate 
z  be  assumed  to  be  of  the  first  order  the  first  two  equations  of  (2)  show  that 
u,  s  contain  in  addition  only  terms  of  the  second  and  higher  orders.  The 
third  equation  of  (2)  has  already  been  considered  in  §  236,  and  when  O  is 
neglected  terms  in  z  of  the  first  order  are  given  by  the  equation 


D*z  =  (22^^)  z. 
Let 

*)  =  S  (n  ~  n')  (t-t2)  =  -  i  log  £2g. 

Then  the  general  solution  is  of  the  form 

iz  =  ak  Sfcf  (£24'+g  -  £-2*-g) 

where  a  preliminary  value  of  g  has  been  found  in  §  240  and  k,  t.2  represent 
the  two  necessary  arbitrary  constants.  As  before  the  suffix  of  £2  has  been 
suppressed  because  it  does  not  affect  the  calculation,  though  the  proper 


246-248]  Lunar  Theory  II  285 

argument  must  be  retained  in  the  results.     The  coefficients  kt  are  deter- 
mined by  equating  terms  in  £2J+e,  so  that 


and  it  is  possible  to  write  k0  =  1. 

In  obtaining  kltk^  to  m2  only  it  is  possible  to  neglect  k2,  7c_2  and  approxi- 
mate values  of  M0,  Ml  =  M^  have  been  found  in  §  240.     Thus  the  equations 

are 

(2  +  g)2^    =  2M0k\ 

(2  -  g)2  k.,  =  2Jtf0&_1 
where 


(2  +  g)s  _  2MQ  =  8,    (2  -  g)2  -  2M0  =  -  4m  -  3m2,   2^  =  2^^  =  »  m2  +  J£  m3 

Hence 

&!  =  T3g  m2,     &_j  =  —  §  m  —  -||  m2 
and  to  this  order  in  m 


iZ  =    ak  {£*  -  £-s  -  (f  m  +  ||  m2 

^  =  2ak  {sin  77  +  (f  m  +  f  f  m2)  sin  (2f  -  77)  +  T3ff  m2  sin  (2£  -1-  77)}. 

248.     Here  the  fundamental  term  is 

z  =  2ak  sin  rj  =  2ak  sin  {g  (n  —  n')  (t  —  t^)} 

and  its  general  meaning  is  easily  seen,  though  the  exact  definition  of  k  must 
be  adapted  to  the  final  approximation  and  then  determined  (like  e)  by  direct 
comparison  with  observation.  The  maximum  value  of  z  is  2ak.  But  it  is 
also  approximately  a  tan  /,  a  being  the  mean  distance  in  the  orbit  projected 
on  the  plane  of  the  ecliptic  and  /  being  the  inclination  of  the  orbit  to  this 
plane.  Hence  k  is  nearly  \  tan  /,  and  differs  little  from  Delaunay's  7=sin  \I. 
Its  provisional  value  may  be  taken  to  be  0'0448866  =  9260". 

At  a  node  2=0  and  the  period  between  successive  returns  to  the  same  node 
is  27r/g(w  —  n').  In  this  time  the  mean  motion  in  longitude  is  27rn/g(n  —  n'). 
Hence  the  mean  rate  of  change  in  the  position  of  the  node  is 

{27rn/g  (n  -  n')  -  2-Tr}  -=-  2?r/g  (n  -  n)  =  n  -  g  (n  -  n') 

=  n  I1  ~  g/C1  +  m)}  =  w  (-  f  m2  +  f|m3) 

with  the  approximate  value  of  g  found  in  §  240.  Since  this  expression  is 
negative  the  lunar  node  has  a  retrogade  motion  and  completes  a  circuit  in 
6890  days  or  18-9  years,  which  is  reduced  by  about  100  days  when  the  com- 
plete value  of  g  is  used.  These  facts  have  an  important  bearing  on  the 
theory  of  eclipse  cycles. 

In  deriving  the  elementary  terms  in  latitude  with  the  characteristic  k  it 
is  enough  to  take  from  the  variational  solution 

r  =  a(l  -  m2cos2£) 
and  to  the  order  m2  the  latitude  is 

z/r  =  2k  (sin  77  +  (f  m  +  ^|m2)  sin  (2£  -  77)  +  |£m2  sin  (2£  +  77)}. 


286  Lunar  Theory  II  [CH.  xxi 

The  first  term,  with  argument  rj  (or  F  in  Delaunay  's  notation)  is  the  principal 
term  in  latitude.  Its  coefficient  is  5°  8'.  The  second  term,  with  argument 
2£  —  77  (or  2D  —  F},  has  been  called  the  evection  in  latitude.  Its  coefficient 
as  found  above  is  610"'6,  the  true  value  being  618"'4.  The  third  term,  with 
argument  2£  -f  77  (or  2.Z)  +  F)  has  the  coefficient  83"'2  as  compared  with  the 
true  value  94"-5. 

249.  It  is  now  possible  to  sketch  the  whole  method  of  the  subsequent 
development.  The  greater  part  of  the  practical  work  of  calculation  has  been 
based  not  on  the  homogeneous  equations  used  above,  which  present  advan- 
tages in  special  cases  (especially  the  calculation  of  long-period  terms),  but  on 
the  original  equations  (2), 

Dhi  +  2mDu  +  f  m2  (u  +  s)  -  --  =  -  d~ 

r*          ds 

KZ  9H 

-  —  =  -!—. 

r3  dz 

It  is  unnecessary  to  use  the  equation  in  s  because  s  =/(£"~1)  if  u  =/(£)  ;  two 
real  equations  are  replaced  by  a  single  complex  one.  Also  the  characteristics 
entering  into  u  and  z  are  distinct.  Hence  the  treatment  of  the  equations  in 
u  and  z  is  also  distinct.  The  order  of  a  characteristic  is  the  sum  of  the 
positive  powers  of  the  parameters  e,  e',  aa'"1,  k  which  compose  it  :  m  is 
a  mere  number  for  this  purpose,  and  retains  its  identity  only  in  the  argu- 
ments. Now  suppose  that  a  complete  solution  u  =  u1,  s  =  sly  z  =  z1  to  the 
order  /*  in  the  characteristics  has  been  obtained.  The  next  step  is  to  find 
the  solution  ic  =  u1  +  uz,  s  =  s1+s.2,  z  —  z1  +  z2,  where  u2,  s.2,  z.2  represent  the 
terms  of  order  fi+1.  Insert  these  values  in  the  equations,  retaining  only 
the  first  powers  of  u2,  s2,  z2.  The  result  is,  since  r2  =  us  +  z*, 

(D  +  m)2  (M!  +  u2)  +  -|-  m2  (^  +  u.2  +  3^  +  3sa)  -  K  (u^  +  u.2)  rrs 

+  ^K'u^r^3  (iiiS2  +  u2si  +  2^i  z2)  =  —  ^ 

OS 

(D2  —  m2)  (Z-L  +  z2)  —  K  (zl  +  z2)  rr3  +  ^icz^-5  (u^  +  u^  +  2^^) 


. 

oz 

Now  terms  of  order  less  than  p  +  1  must  be  satisfied  identically  and  therefore 
terms  linear  in  ult  sl}  zl  may  be  omitted.  Also  terms  of  order  higher  than 
/4+  1  can  be  neglected.  Hence  uly  s1}  z^  may  be  used  in  calculating  H,  and 
in  conjunction  with  u2,  s2,  z2  it  is  possible  to  write  U^  —  UQ,  s1  =  s0,  ^  =  0, 
ri*  =  UQSO  =  p<?,  where  u0)  s0>  z=0  is  the  variational  solution  of  zero  order. 
Hence  the  equations  reduce  to 
(D  +  m)2  1/2  +  u2  (|m2  +  ^Kp0~3)+  s2  (f  m2  +  f  KU<?p0-r') 


-  z,  (m2 


248-25i]  .  Lunar  Theory  II  287 

where  the  terms  with  D  have  been  retained  on  the  right-hand  side,  though 
apparently  of  order  not  higher  than  /*,  for  a  reason  to  be  explained  later. 
For  the  moment  they  can  be  left  out  of  sight. 

250.  Since  the  treatment  of  the  two  equations  is  separate  but  quite 
similar  it  will  be  enough  to  consider  the  first.  It  is  convenient  to  write 
Ui  =  M0  +  Ui,  s1  =  s0  +  Si  and  to  expand  the  term  KU^T^  in  terms  of  w/,  s/,  z1} 
rejecting  the  variational  part  KU0p0~3  and  the  linear  terms.  The  form  of 
the  known  solution  has  been  made  sufficiently  obvious,  and  it  is  clear  that 
the  right-hand  side,  when  developed,  will  contain  an  aggregate  of  character- 
istics A  each  of  order  //.  +  1  and  each  associated  with  one  or  more  series, 
Each  constituent  part  may  be  taken  to  be  of  the  form 
A  =  aA2  A{*»  +  A'  -4-** 


where 

r  =  q1c  +  qam  +  q3g 

q\,  q-2,  q*  having  fixed  integral  values  (positive  or  negative)  in  the  series  con- 
sidered, while  2i  may  have  odd  integral  values  when  aa'"1  occurs  in  A.. 

The  part  of  the  solution  required  to  satisfy  this  series  is  of  the  same  form 


and  \,  A,/  are  to  be  found  by  inserting  this  expression  in  the  equation.     This 
may  be  written 

(D  +  m)2  uz  +  Muz  +  Ns&=  A 
where 

M  =    m2  +    /c0-3  =2*      N    =    m2  +    KU*-*= 


The  series  M,  in  which  Mi  =  M^,  has  already  occurred  in  the  determination 
of  c0  and  g0.  After  substitution  of  the  series  for  u2,  s2  comparison  of  the 
terms  in  £>±<2./+T>+1  on  both  sides  of  the  equation  gives 

'i-j  =  Aj    } 

This  series  of  linear  equations,  in  which  the  coefficients  Mi}  Nf  rapidly  diminish, 
must  then  be  solved  by  successive  approximation.  When  this  has  been 
carried  out  for  each  series  A  and  every  characteristic  A,  all  the  terms  of  order 
fjL  +  1  in  u,  s  have  been  determined.  The  treatment  of  z  is  precisely  similar. 

251.  But  one  important  question  clearly  arises.  Is  the  set  of  linear 
equations  consistent  and  definite  ?  If  the  modulus  of  the  set,  which  can  be 
written  as  a  symmetrical  determinant  of  infinite  order  since  Mt  =  M_i} 
Ni  =  N-i,  is  not  zero,  the  solution  is  certainly  definite.  This  is  the  general 
case.  But  consider  the  determination  of  e^,  e/  the  co-factors  of  the  character- 
istic e  of  the  first  order.  By  the  above  method  these  will  be  obtained  from 
(23)  by  putting  Aj  =  A'  ^  =  0  and  r  =  c.  The  consistency  of  the  equations 


Lunar  Theory  II  [CH.  xxi 

now  requires  the  modulus  to  vanish.  It  is  obvious  that  this  condition  in  fact 
musfc  lead  to  a  determination  of  r  which  will  be  identical  with  the  value  of 
c0,  though  the  latter  was  found  above  in  a  formally  different  way.  When 
the  equations  have  thus  been  made  consistent  the  solution  only  becomes 
definite  when  the  arbitrary  condition  e0  -  e0'  =  1  is  added,  and  this  condition 
is  equivalent  to  a  definition  of  e. 

It  is.  now  evident  that  the  modulus  vanishes  whenever  r  =  c,  or  for  every 
series  based  on  the  same  argument  as  that  of  the  principal  elliptic  term. 
The  consistency  of  the  linear  equations  requires  a  relation  between  the 
coefficients  Aj}  A]  which  may  be  expressed  by  equating  the  modulus  to  zero 
after  replacing  any  column  in  it  by  the  series  Ah  A-.  But  owing  to  the 
symmetry  of  the  modulus  this  relation  is  capable  of  a  much  simpler  form. 
Let  the  equations  (23)  be  multiplied  by  e/,  e'_,-  and  let  the  sum  be  taken  for 
all  values  of  j.  Then  the  coefficient  of  X,-  is 

(2j  +  r  +  1  +  m)2  e,-  +  2Mi€j+i  +  2  #»•€_,•+<  =  0 


because,  since  ^Mi€j+i=^M_iej^i  =  ^Miej,i)  this  is  one  of  the  equations  of 
condition.  Similarly  all  the  coefficients  on  the  left-hand  side  vanish,  and 
the  required  relation  appears  in  the  form 

+  A'_je'-j)  ...........................  (24) 


j 


The  reason  for  retaining  the  terms  (D2  +  2mZ))z«i  in  (22)  will  now  be  under- 
stood. Without  them  there  is  no  reason  why  the  relation  (24)  should  be 
satisfied,  and  in  fact  it  will  be  contradicted.  But  let  wx  contain  terms  of  the 
form 


&  {[c2  +  2c  (2t  +  1  +  m)]  #^+c 

+  [c2  +  2c  (2i  -  1  -  m)]  E'-g-*^} 

where  terms  obviously  of  order  less  than  //,  +  1  are  omitted.  Then  clearly,  if 
the  value  of  c  here  be  regarded  as  unknown,  it  will  be  possible  to  adjust  its 
value  so  as  to  satisfy  the  relation  (24). 

252.  The  matter  is  made  clearer  by  considering  the  actual  facts.  In  the 
first  order  there  is  one  such  series,  with  the  coefficients  ei}  e/.  In  the  second 
order  there  is  no  such  series  and  the  question  does  not  arise.  The  primitive 
value  c0  suffices.  In  the  third  order  series  of  this  type  reappear,  associated 
with  the  characteristics  e3,  ee'2,  ek2,  e  (a  a'"1)2.  The  contemplated  change  in  c 
is  associated  with  e  through  the  first  order  terms.  Hence  the  relation  (24) 
in  the  third  order  will  give  in  succession  the  parts  of  c  which  contain 
e2,  e'2,  k2  and  (aa'"1)2.  Similarly  still  higher  parts  of  c  may  be  found  in  con- 
junction with  the  inequalities  of  a  higher  order.  It  is  natural  that  the 
motion  of  the  perigee  (and  the  value  of  the  characteristic  exponent)  which 
was  determined  for  highly  simplified  conditions,  should  require  adjustment 


25i-i>53]  Lunar  Theory  II  289 

when  the  conditions  are  more  complicated  and  the  deviation  from  the  periodic 
orbit  is  no  longer  infinitely  small. 

For  c  let  Cj  +  X'Sc  be  written,  where  X'Sc  is  the  part  to  be  determined,  its 
characteristic  being  X',  and  let 


where  Bj,  B'_j,  Dj,  D'_,  are  calculated  numbers.  With  the  new  value  of  c  the 
quantities  Aj,  A'  _-t  satisfy  a  certain  relation  identically  as  required,  and  the 
equations  (23)  become  consistent,  but  the  solution  is  not  definite  because  any 
one  of  the  equations  can  be  derived  from  the  rest.  An  arbitrary  condition 
can  be  imposed,  and  the  form  X0'  =  X0  is  chosen.  The  solution  is  then  con- 
ducted in  the  following  way. 

The  equations  for  j  =  0  are  left  aside.  Three  separate  solutions  are  then 
made  of  the  remaining  equations:  (1)  \j  =  bj,  X'_y  =  6'_,-  when  \0  =  X0'  =  0 
and  Aj  =  Bj,  A'-}  =  B'_}  ;  (2)  X;  =  djt  \'_,-  =  d'_j  when  X0  =  X0'=0  and  Aj=Dj, 
A'_}  =  iy_j  ;  and  (3)  X,-  =fjt  X'_;  =  /'_/  when  X0  =  X0'  =  1  and  A}  =  A'  _}  =  0. 
The  last,  which  under  the  different  condition  X0  —  X0'  =  1  would  have  led  to 
€j,  e'_j,  is  independent  of  Aj,  A'^  and  applies  in  all  cases.  The  complete 
solution  is  therefore 

\.  =  bj  +  dj  Sc  +fj\,     \'-j  =  b'-j  +  d'-j  Sc  +/'_,•  X0  . 
When  these  are  inserted  in  the  equations  for  j  =  0  the  result  is  of  the  form 

b,  +  d0Sc  +/0X0  =  &„'  +  d0'Sc  +/0'X0  =  0 

and  Sc  and  X0  are  thus  determined.  The  value  of  Sc  must  also  satisfy  the 
relation  (24),  so  that  a  check  on  the  accuracy  of  the  work  is  provided.  The 
solution  of  the  equations  (23)  for  the  case  when  r  =  c  is  therefore  complete, 
and  the  derivation  of  the  higher  parts  of  c  has  been  explained.  It  may  be 
noted  that  on  the  left-hand  side  of  these  equations  the  primitive  value  c0  is 
to  be  retained  for  r  at  every  stage,  both  because  it  is  associated  with  terms  of 
the  full  order  yu,  +  1  and  because  the  theory  of  the  equations  depends  on  the 
fact  that  the  modulus  vanishes.  On  the  other  side  c  will  receive  its  full 
value  so  far  as  it  has  been  determined.  When  a  new  part  of  c  comes  to  be 
determined  in  conjunction  with  inequalities  having  the  characteristic  X,  8c  is 
always  associated  through  (D2  -I-  2mD)  (MJ)  with  the  terms  in  i^  of  the  first 
order  in  e.  Hence  the  new  part  of  c  itself  always  has  the  characteristic 
X'  =  e~xX,  and  the  numbers  dj,  d'_j,  like/-,/'-/,  are  the  same  in  all  cases. 

253.  With  the  equation  for  z  matters  follow  a  precisely  similar  course, 
and  the  exceptional  case  arises  when  r  =  g.  The  conditions  are  simpler, 
because  X,-  +  X'_,-  =  0  always,  and  therefore  the  arbitrary  relation  has  the 
form  X0  =  V  =  0.  The  terms  of  the  first  order  with  suitable  arguments  have 
the  characteristic  k,  and  the  part  of  g  found  in  conjunction  with  inequalities 
having  the  characteristic  X  contains  the  characteristic  k-1X. 

P.  D.  A. 


290  Lunar  Theory  II  [CH.  xxi 

The  arbitrary  condition  X0  =  X0'  adopted  in  all  cases  has  an  importance 
beyond  that  apparent  in  the  actual  calculation.  The  aggregate  of  the  terms 
considered  up  to  the  final  stage  of  approximation  gives  for  the  one  argument 


u  =  ae 

*  =  ae 


The  last  expression  remains  unaltered  throughout  the  course  of  the  approxi- 
mations. Hence  the  constant  e  is  defined  as  "  the  coefficient  of  a  sin  I  in 
the  final  expression  of  p  sin  (v  —  nt  —  e)  as  a  sum  of  periodic  terms,  where 
v  —  nt  —  e  is  the  difference  of  the  true  and  mean  longitudes  and  p  is  the 
projection  of  the  Moon's  radius  vector  on  the  plane  of  reference." 

Similarly  the  terms  of  the  form 


in  the  first  approximation  have  no  addition  made  to  them  subsequently, 
since  X0  =  X/  =  0.  Hence  the  constant  k  is  defined  as  "  the  coefficient  of 
2a  sin  F  in  the  (final)  expression  of  z  as  a  sum  of  periodic  terms." 

There  is  no  reason  to  alter  the  definition  of  a,  which  is  based  on  the 
variational  curve.  But  it  is  then  to  be  noticed  that  the  constant  of  distance 
in  the  projection  on  the  z  plane  will  no  longer  be  aa0,  where  a0  =  1,  but  will 
be  affected  by  terms  with  various  characteristics  which  arise  in  the  course  of 
the  approximations  as  the  constant  parts  of  u%~1  or  s£  Either  m  or  a,  since 
they  are  connected  by  a  certain  relation  (11),  maybe  regarded  as  an  arbitrary 
constant  of  the  solution. 

The  remaining  three  arbitraries  have  been  denoted  by  t0,  tlf  t2.  These 
may  be  replaced  by  e,  -or,  6,  the  mean  longitudes  of  the  Moon  and  its  perigee 
and  node  at  the  epoch  t  =  0.  Then 

D  =      (n  —  n')  (t  -  t0)  =     (n  -n')t+e-  e' 
I    =  c  (n  —  n)  (t  —  ti)  =  c  (n  —  ri)  t  +  e  —  OT 
I'  =m(n-  n')  (t  -  t3)  =     n't  +  e  -  -er 
F  =  g  (n  -  n')  (t  -  t.,)  =  g  (n  -  n'}  t  +  e-0 

where  e'  is  the  mean  longitude  of  the  Sun  at  the  epoch  t  =  0  and  vr'  is  the 
(constant)  longitude  of  the  solar  perigee.  The  time  ts  is  not  an  arbitrary  :  it 
depends  on  the  Sun  alone  and  is  one  of  the  data  of  the  problem. 

The  formulae  for  transformation  to  polar  coordinates  were  given  in  §  230 
for  two  dimensions  only.  It  is  necessary  to  replace  r  by  p,  its  projection  on 
the  plane  of  the  ecliptic,  where  p2  =  X2  +  Y2  =  us.  Then 

u%~1  =  p  exp.  i(v  —  nt  —  e) 
s£      =  p  exp.  —  i  (v  —  nt  —  e) 
z       =  p  tan  <f> 


253,  254]  Lunar  Theory  II  291 

where  <f>  is  the  latitude.     Hence  the  true  longitude  and  the  latitude  are 


v  =  nt  +  e 


p    p 

The  constant  of  the  Moon's  horizontal  equatorial  parallax  is  based  on  a, 
where  n2a?  =  E  +  M.  To  obtain  the  parallax  at  any  time  this  constant  must 
be  multiplied  by 

ns 
a2 


a     i 
=  s.'  V 


In  these  expressions  for  v,  <f>  and  ar~l  the  variational  parts  u0,  s0  are  separated 
from  the  other  terms  MI}  slt  z,  and  the  expressions  are  then  expanded  in  terms 
of  the  latter.  Advantage  can  thus  be  taken  of  the  expansions  already  obtained 
in  the  course  of  the  previous  work.  The  conversion  to  the  final  form  of 
coordinates  therefore  entails  no  great  amount  of  extra  labour. 

254.  This  completes  in  outline  the  solution  of  the  main  part  of  the 
problem,  in  which  the  Earth,  Moon  and  Sun  are  treated  as  centrobaric 
bodies,  and  the  orbit  of  the  Sun,  or  the  relative  orbit  of  the  centre  of  mass 
of  the  Earth-Moon  system,  is  treated  as  an  undisturbed  ellipse  in  a  fixed 
plane.  A  large  number  of  comparatively  small  but  highly  complicated 
corrections  are  still  necessary  in  order  to  represent  the  gravitational  motion 
of  the  Moon  in  actual  circumstances.  They  may  be  classified  thus  : 

(1)  The  effect  of  the  ellipsoidal  figure  of  the  Earth,  and  possibly  of  the 
Moon. 

(2)  The  direct  action  of  the  planets  on  the  relative  motion  of  the  Moon. 

(3)  The  indirect  action  of  the  planets,  which  operates  by  modifying  the 
coordinates  of  the  Sun.     These  indirect  effects  are  in  general  larger  than 
the  direct  effects,  and  are  sometimes  sensible  in  the  lunar  motion  when  they 
are  insensible  in  the  relative  motion  of  the  Earth  and  Sun.     Among  the 
indirect  actions  of  the  planets  may  be  specially  mentioned 

(4)  Lunar  inequalities  produced  by  the  motion  of  the  ecliptic,  and 

(5)  The  secular  acceleration  of  the  Moon's  mean  motion,  which  arises 
from  the  secular  change  in  the  solar  eccentricity  e  under  the  action  of  the 
planets. 

It  is  impossible  to  discuss  these  matters  profitably  in  a  short  space.  The 
reader  will  find  references  in  Professor  Brown's  Treatise  and  detailed  results 
in  the  memoir*  which  contains  his  complete  and  original  theory. 

*  Memoirs  R.  Astr.  Soc.,  MIJ,  pp.  39,  Ifi3  ;  LIV,  p.  1  ;  LVII,  p.  51  ;  LIX,  p.  1. 

19—2 


CHAPTER  XXII 

PRECESSION,    NUTATION    AND    TIME 

255.  In  order  to  investigate  the  motion  of  the  Earth  about  its  centre  of 
gravity  0  we  take  a  set  of  rectangular  axes  OXYZ  fixed  in  space  and  a 
second  set  Oxyz  coinciding  with  the  principal  axes  of  inertia.  These  are 
fixed  in  the  Earth  and  move  with  it.  The  two  sets  are  drawn  in  such  a 
sense  that  the  positive  directions  of  the  corresponding  axes  can  be  brought 
into  coincidence  by  a  suitable  rotation.  Their  relative  situation  is  defined 
by  the  three  Eulerian  angles  6,  </>,  ty,  where  6  is  the  angle  betwreen  OZ 
and  Oz,  <£  is  the  angle  between  the  planes  OXZ  and  OZz,  and  ty  is  the  angle 
between  the  planes  OZz  and  Ozx.  Then  the  coordinates  are  related  by  the 
scheme  : 

X  Y  Z 

x      cos  9  cos  0  cos  i/r  —  sin  <f>  sin  -\Jr       cos^sin^cos^-f  cos<£sini|r    —  sin#cos-*/r 

y  —  cos  6  cos  <f)  sin  i|r  —  sin  <f>  cos  -fy  —  cos  #  sin  $  si  n-\Jr  + cos  <£  cosier       sin  0  sin  ty 
z  sin  6  cos  <£  sin  6  sin  </>  cos  6 

The  result  of  resolving  the  angular  velocities  6  which  is  a  rotation  in  the 
plane  OZz,  <£  which  is  a  rotation  about  OZ,  and  -^  which  is  a  rotation  about 
Oz,  about  Ox,  Oy,  Oz  is  to  give  the  equivalent  angular  velocities  about  these 
axes,  namely 

twj  =  6  sin  i/r  —  <f)  sin  6  cos  ty 

G>2  =  0  cos  ^  +  <j>  sin  6  sin  i/r       (1) 

&>3  =  yjr  +  <£  cos  $ 

which  are  Euler's  geometrical  equations. 

Let  A,  B,  C  be  the  moments  of  inertia  about  the  axes  Oxyz  and  L,  M,  N 
the  moments  of  the  external  forces  about  these  axes.  Then  the  dynamical 
equations  may  be  written  in  the  well-known  form : 

/  D      /"»\  r 

~  I  O  —   U  )  60.2  61)3  :=  ±J 

«2  -(C-^)ws«i  =  ^(-    (2) 


255,  256]  Precession,  Nutation  and  Time  293 

256.  The  external  forces  which  are  here  considered  are  due  to  the  action 
of  the  Sun  and  Moon.  An  approximate  expression  for  the  action  of  either 
of  these  bodies  is  sufficient  and  easily  found.  The  potential  of  the  Earth 
(mass  m)  at  a  distant  point  P  has  been  found  (§  18)  to  be 


T7.     n^  dm     n  (m 

V  =  G-2,  --    =  tr 


, 

1 


p  \r  2?'3 

where  OP  =  r  and  /  is  the  moment  of  inertia  of  m  about  OP.  This  expression 
is  true  as  regards  terms  of  the  second  order  in  the  coordinates  of  points  in  m 
relative  to  the  centre  of  gravity  0.  Terms  of  the  third  order  will  clearly 
vanish  in  the  sum  provided  that  the  mass  m  possesses  three  rectangular 
planes  of  symmetry  :  and  this  is  sensibly  true  in  the  case  of  the  Earth. 
Terms  of  the  fourth  order  are  small  in  consequence  of  the  ellipsoidal  figure 
of  the  Earth  and  are  neglected.  Now  V  is  the  work  done  by  unit  attracting 
mass  at  P  when  the  particles  of  the  mass  m  are  brought  from  infinity  to 
their  actual  configuration.  Hence  the  work  done  by  a  finite  mass  near 
a  distant  point  0'  is 


by  similar  reasoning,  if  0'  is  the  centre  of  gravity  of  the  attracting  mass 
m,  00'  =  R,  A',  B',  C'  are  the  principal  moments  of  inertia  of  m'  at  0'  and  /' 
is  the  moment  of  inertia  of  m'  about  00'.  Now  since  A,  B,  C  and  /  are  of 
the  second  order  in  the  linear  dimensions  of  m,  terms  of  the  second  order  in 
the  linear  dimensions  of  m'  can  be  neglected  when  associated  with  them. 
Let  the  coordinates  of  0'  relative  to  0  be  (as,  y,  z)  and  of  P  relative  to  0'  be 
1,-  Then 


7-2/  =  A  (x  +  £f 

But  since  0'  is  the  centre  of  gravity  of  the  mass  m' 
2,!;  dm'  =  2,r)dmf  =  ^dm  =  0. 

Hence  if  the  expression  to  be  summed  be  expanded  in  terms  of  £,  ?;,  £  the 
terms  of  the  first  order  vanish  in  the  sum  and  terms  of  the  second  order  are 
neglected.  To  this  order  of  approximation 

A+B+C     3  (An?  +  By*  + 
-  -- 


and  if  /  now  represents  the  moment  of  inertia  of  m  about  00',  the  complete 
expression  for  U  becomes 

f  mm'     m  (A'  +  B'  +  C'-M')     m'  (A  +B  +  G-  31)] 
r(R  2R3  2R3  }' 


294  Precession,  Nutation  and  Time  [CH.  xxn 

This  represents  the  mutual  potential  of  two  masses  m,  m  with  sufficient 
accuracy.  In  the  usual  astronomical  units  (§  24)  G  =  A:2.  The  mass  of  the 
Sun  is  unity  and  for  the  masses  of  the  Earth  and  Moon  we  take  E  and//?. 
Then  if  the  mean  distances  of  the  Sun  and  Moon  are  a'  (=  1  )  and  a"  and  the 
mean  motions  n'  and  n"  '  , 

Gl+E     =  ri*a's 


257.  The  moments  of  the  external  forces  about  the  axes  Oxyz  being 
L,  M,  N,  the  work  done  by  them  when  the  Earth  receives  a  small  twist 
defined  by  the  rotations  da)1}  da)2,  da)3  about  the  same  axes  is 

dU=L  dwl  +  Md(o.2  +  Nd(os  . 

But  U  depends  on  the  orientation  of  the  Earth  only  through  the  occurrence 
of  /  ;  and 

R2I  =  Ax*  +  By*  +  Cz* 

(x,  y,  z)  being  the  centre  of  gravity  of  the  attracting  body.     Hence 

dU=-  3Gm  (Ax  dx  +  Bydy  +  Czdz)/R5. 
But  with  due  regard  to  sign,  when  the  axes  are  rotated, 

dx  =  y  d(D3  —  z  da)2,     dy  =  zdwl  —  xdw.A,     dz  =  xda)2  —  ydwl. 

Hence,  equating  the  coefficients  of  d(ol,  dw2,  dws  in  the  two  expressions 
fordU, 

L  =  3Gm'(C-B}yzlR5,    M  =  3Gm'(A  -  C)xz\R\    N  =  3GW  '(B-  A)xyj  R5. 

These  apply  to  a  body  possessing  three  distinct  principal  axes.  But  the 
Earth  may  be  regarded  as  an  ellipsoid  of  revolution,  for  which  E  —  A  and 
C>A.  Under  these  circumstances 

L  =  SGm'(C-  A)yz/R5,     M=  -  3GW  (G  -  A)  xzjR',    N  =  Q. 

On  the  other  hand,  the  term  in  U  which  depends  on  the  orientation  of  the 
Earth  is  more  generally 


'  {(20-  A  -B)z*  +  (A-  B)  O2  -  ?/2)  +  (A+B)  R>]  /R5 

a  useful  form  for  some  purposes.  The  last  term  on  the  right,  being  inde- 
pendent of  the  orientation,  can  always  be  rejected  ;  and  when  the  Earth 
is  considered  uniaxal,  it  is  possible  to  use  simply 

U"  =  -§Gm'(C-A)z*IR>  ........................  (3) 

258.     With  B  =  A  and  N=  0,  the  third  equation  of  (2)  gives 

&>3  =  0,  <u3  =  n 
and  the  other  equations  of  the  set  become 

J.&)!  +  (C  —  A)  na)2  =  L 
Aw2  —  (C  —  A)  nwv  =  M. 


256-259]  Precession,  Nutation  and  Time  295 

The  actual  motion  of  the  Earth  is  a  steady  state  of  rotation  disturbed  by  the 
external  forces  and  this  steady  state  will  be  found  by  putting  L  —  M  =  0. 
The  equations  then  give 

&>!  +  fJ?(t)\  —  d>2  +  fJ?W.2  =  0 

where 


Hence  the  steady  state  is  given  by 

(ol  —  h  cos  (p,t  +  a),     o).2  =  h  sin  (/mt  +  a). 
But  the  instantaneous  axis  of  rotation  in  the  Earth  is  the  line 

#/o>i  =  y/e>a  =  zf(D3 
or 

ac/h  cos  (pt  +  a)  =  y/h  sin  (/u.t  +  a)  =  z\n 

which  indicates  that  if  h  is  fairly  small  the  terrestrial  pole  describes  a  small 
circle  of  radius  h/n  about  the  axis  of  figure  in  the  period  2ir//j,.  This  is  the 
Eulerian  period  of  A/(C—  A)  (roughly  300)  days.  Now  the  angle  between 
the  Zenith  of  a  place  and  the  Pole  is  the  co-latitude  of  the  place,  an  angle 
which  can  be  constantly  observed.  Hence  the  latitude  of  any  place  should 
exhibit  a  variation  with  a  period  of  about  10  months.  Until  a  quarter  of 
a  century  ago  no  variation  of  latitude  had  certainly  been  detected.  Since 
that  time  variations  (of  the  order  of  0"'3)  have  been  systematically  observed 
and  studied  and  have  also  been  traced  in  the  older  observations.  But 
analysis  has  proved  conclusively  that  these  variations  contain  no  part  which 
conforms  with  the  Eulerian  period.  They  cannot  therefore  be  explained  by 
the  free  motion  of  the  Pole  on  a  rigid  Earth.  Hence  observation  justifies 
the  belief  that  h/n  is  insensibly  small. 

The  variations  of  latitude  observed  are  always  very  small  and  constitute 
a  highly  complex  phenomenon.  The  periods  of  the  chief  components  of  the 
motion  of  the  Pole  are  about  12  and  14  months. 

259.  Corresponding  to  the  free  movement  of  the  Pole  on  the  Earth's 
surface  we  have,  by  (1), 

6  =  d)1  sin  T/T  +  (0.2  cos  ty  =     h  sin  (/*£  +  a  +  i|r) 
$  sin  6  =  coo  sin  ty  —  a>}  cos  i/r  =  —  h  cos  (/j,t  +  a  +  ty). 

For  the  plane  OXY  we  take  the  plane  of  the  ecliptic  which  varies  but 
slightly  in  consequence  of  planetary  perturbations.  The  value  of  6  is  about 
23°.  Hence  6  and  <£  are  very  small  in  comparison  with  n,  a  fact  in  accord- 
ance with  observation  even  when  the  disturbing  effects  of  the  Sun  and 
Moon  are  operative.  Hence,  further,  -^  differs  only  slightly  from  n. 

The  rotational  energy  of  the  Earth  is  T,  where 
2T=^(<o12  +  <y22)  +  CW 

=  A  (fc  +  <j>2  sin2  0)  +  C  (ijr  +  <j>  cos  0)2. 


296  Precession,  Nutation  and  Time  [en.  xxn 

Hence  the  Lagrangian  equations  of  motion  are 

-r  (A6)  -  Aft  sin  0  cos  0  +  Cty  sin  0  (-^  +  </>  cos  0)  =  — 


But  since 

T  =  JV  =  0,  T^  +  <J)  cos  0  =  n 
dty 

the  first  two  equations  become 

f)TJ~ 
A  6  —  A  (f>2  sin  6  cos  6  +  Cn  6  sin  6  =  — 

0(7 

ft  rlTT 

~(A(f>  sin2  B  +  Cn  cos  0)  =  |^  . 

a£  c<£ 

It  has  been  seen  that  n  is  very  large  compared  with  0  and  <£,  and  it  follows 
that  those  terms  are  of  predominant  importance  which  contain  n  as  a  factor. 
Neglecting  the  other  terms  on  the  left  the  equations  become  simply 

.  1       3U 


0= 


(7n  sin  0 


The  complete  justification  for  omitting  the  terms  rejected  must  be  sought 
by  substituting  in  them  the  results  which  follow  from  the  latter  simple  form 
of  equations,  when  it  will  be  found  that  they  are  practically  insensible.  The 
form  to  be  used  for  U  is  given  by  (3),  so  that 


a  sum  of  two  terms  corresponding  to  the  Sun  and  Moon.  For  each  dis- 
turbing body  it  is  necessary  to  find  the  product  of  z2jR2  and  a3/R3  expressed 
in  appropriate  terms  and  with  a  suitable  degree  of  approximation. 

260.  The  axes  XYZ  being  fixed  in  space  are  defined  so  that  OZ  is 
directed  towards  the  pole  of  the  ecliptic  for  1850.0  and  OX  towards  the 
equinox  for  the  same  epoch.  By  the  scheme  of  transformation 

z  =  X  sin  0  cos  <f)  +  Fsin  0  sin  <f)  +  Z  cos  0. 

The  position  of  a  disturbing  body,  such  as  the  Moon,  is  more  conveniently 
referred  to  a  similar  set  of  axes  for  another  epoch  t.  The  necessary  changes 
may  be  considered  successively,  thus  : 


2r>9,  260]  Precession,  Nutation  and  Time  297 

(i)     Rotate  the  axes  about  OZ  through  the  angle  fl  so  as  to  bring  OX  to 
the  position  OX^.     Then 

X  =  X,  cos  O  -  Fa  sin  fl,     Y  =  Yl  cos  fl  +  J^  sin  fl,     Z  =  Zl 

where  fl  is  the  node  of  the  ecliptic  for  epoch  t  on  the  ecliptic  for  1850.0. 

(ii)     Rotate  the  axes  about  OX^  through  the  angle  i  so  as  to  bring  OYl 
to  the  position  OF,.     Then 

X1  =  X2,     Fj  =  F2  cos  i  —  Z2  sin  i,     Zl  =  Z2  cos  i  +  F2  sin  i 
where  i  is  the  inclination  of  the  ecliptic  for  epoch  t  to  the  ecliptic  for  1850.0. 

(iii)     Rotate  the  axes  about  OZ2  through  the  angle  N  —  fl  so  as  to  bring 
OX. 2  to  the  position  OX3.     Then 

X2  =  X3  cos  (N-  fl)  -  F3  sin  (N  -  fl), 

F2  =  F3  cos  (#  -  fl)  -f  X3  sin  (>V  -  fl),     Z,  =  Z, 

where  N  is  the  longitude  of  the  Moon's  node  reckoned  through  fl  in  both 
ecliptic  planes. 

(iv)     Rotate  the  axes  about  OX3  through  the  angle  c  so  as  to  bring  OF3 
to  the  position  OY4.     Then 

X3  =  X4,     Y3  =  F4  cos  c  —  Z4  sin  c,     Z3  =  Z4  cos  c  +  F4  sin  c 
where  c  is  the  inclination  of  the  Moon's  orbit  to  the  ecliptic  for  epoch  t. 
But,  if  (X4,  F4,  Zt)  are  the  Moon's  coordinates, 

X4  =  r  cos  (v  -  N),     Y^  =  r  sin (v  ~  N),     Z4=0 

where  r  is  the  radius  vector  and  v  is  the  longitude  of  the  Moon  at  epoch  t 
reckoned  in  its  orbit;  this  longitude  is  the  sum  of  three  arcs  in  the  two 
ecliptic  planes  and  the  plane  of  the  lunar  orbit.  Now  i  <  1°  and,  for  the 
Moon,  c  is  of  the  order  5°.  Terms  of  the  order  t2,  c3  and  ic  are  therefore 
neglected.  Then  the  result  of  eliminating  (X3,  Y3,  Z3),  (X4,  Y4,  Z4)  gives 

X.2  =  r  cos  (v  -  fl)  +  |c2r  sin  (v  -  N)  sin  (N  -  fl) 
F2  =  r  sin  (v  -  fl)  -  |c2r  sin  (v  -  N)  cos  (N  -  fl) 
Z2  =  cr  sin  (v  —  N) 

arid  the  result  of  eliminating  (X,  F,  Z),  (Xlt  F1}  ZJ  gives 
0  =  X2  sin  0  cos  ((/>  -  fl)  +  F2  sin  0  sin  (0  —  fl)  +  ^2  cos  0 

+  i  { Y2  cos  8  -  Z2  sin  0  sin  (</>  -  fl)}. 
Hence 

z/r  =  sin  6  cos  (v  —  </>)  +  c  cos  6  sin  (v  —  JV)  —  |c2  sin  0  sin  (v  —  N)  sin  (0  —  JV) 

+  *  cos  6  sin  (v  —  fl). 

In  squaring  this  expression  terms  not  involving  6  or  (f)  can  be  rejected, 
because  they  disappear  on  differentiation.  Also  terms  involving  v  with 


298  Precession,  Nutation  and  Time  [on.  xxn 

coefficients  above  zero  order  are  found  to  be  negligible  in  effect.  Under 
these  conditions  the  result  becomes 

^/r2  =  £  Sin2  0  +  £  Sin2  Q  cos  2  (v  -  </>) 

+  c  sin  0  cos  6  sin  (<f>  —  N)  +  i  sin  0  cos  #  sin  (</>  —  H) 

+  |c2sin20cos2(<£-7\0-fc2sin20  (4) 

261.  Certain  expansions  in  terms  of  the  mean  anomaly  in  undisturbed 
elliptic  motion  are  now  required.  When  es  is  neglected  in  the  formulae 
of  §  40,  (22),  (26)  and  (27)  of  Chapter  IV  become 

r/a  =  1  +  |e2  -  e  cos  M  -  %e2  cos  2M 
tfxlr3  =  (1  -  f  e2)  cos  M  +  2e  cos  23f  +  %7-e2  cos  3M 
tfy/r3  =  (1  -  f  e2)  sin  .¥  +  2e  sin  2J/  +  ^7-e2  sin  3M. 
The  latter  give,  w  being  the  true  anomaly, 

a4 sin  2w/r4  =  (1  -  e2)  sin  2Jlf  +  4e  sin  3if  +  -\3-e2  sin  4<M 

a4 cos  2ry/r4  =  £e2  +  (1  -  e2)  cos  271/7  +  4e  cos  3Jf  +  ^3-e2  cos  4if 

a4/?^4  =  1  +  3e2  +  4e  cos  Jlf  +  7e2  cos  2M 
whence,  after  multiplication  by  r/a, 

a3  sin  2w/r3  =[-^e  sin  J/]  +  (1  -  f  e2)  sin  2M  +  [f  e  sin  3J/  +  -1/ e2  sin 
a3  cos  2w/r3  =  [-  \e  cos  M]  +  (1  -  f  e2)  cos  2M  +  [$e  cos  3if 
a»/r*  =  1  +  f  e2  +  3e  cos  M  +  [f  e2  cos  2M]. 

The  eccentricity  being  small,  of  the  same  order  as  c,  the  terms  [  ]  which 
involve  M  and  are  not  of  zero  order,  are  immediately  rejected.  Now 

M  =  ri't  +  /i  —  OT 
v   =w  +  & 

where  n't  +  /A  is  the  mean  longitude  of  the  Moon  in  its  orbit  and  -57  is  the 
longitude  of  the  lunar  perigee,  both  being  measured  partly  in  the  two 
ecliptic  planes  for  185OO  and  the  epoch  t  and  partly  in  the  plane  of  the 
lunar  orbit.  From  the  expression  (4)  can  now  be  derived 

a3*2  /r5  =  (i  -  f  c2  +  f  e2)  sin2  6  +  c  sin  0  cos  6  sin  (<f>  -  N) 
+  i  sin  0  cos  0  sin  (<£  -  fl)  +  £c2 sin2  6  cos  2(<j>-N) 
+  %  sin2  6  cos  2  (n"<  +  n  -  <£)  +  f  e  sin2  0  cos  (»"£  +  /*  -  «r) 
the  final  term  being  retained  though  periodic  and  not  of  zero  order. 
For  the  Sun  c  =  0  and  hence  similarly 
a'V2//8  =  (|  +  f  e'2)  sin2  0  +  *'  sin  6  cos  0  sin  (</>  -  fl) 

+  i  sin2  6  cos  2  (w'$  +  //  -<#>)  +  f  e'  sin2  0  cos  (n'£  +  //  -  ™'}. 


260-263]  Precession,  Nutation  and  Time  299 

262.     These  expressions  give  the  means  of  forming  U,  for 

For  the  Moon  (§  256) 


<  =GEf  =  fn,"* 
and  for  the  Sun 

a-'"'  =GL=  jrc'2_ 

a3       a'3     1  +  E ' 
Let 

C-A    >"<  C-A      £ 

A8~¥'~ar-i+/'     Kl-*-~CT'\^E    (5 

Then 

U  _  a?zz  a'sz'2 

~lr<r   —  "~~  -**-2  •          iT~  ~~  •"•!  •          >v 

Cn  r5  r5 

=  -  [K,  (£  -  I  c2  +  |  e2)  +  K!  (£  +  f  e'2)}  sin2  0  -  £(^i  +  Kz)  i  sin  26*  sin  (0  -  fl 

-  J&LJ  {£  cos  2  (w'£  +  //  —  <£)  +  f  e'  cos  (n'£  +  //  —  OT')}  sin2  0 

-  iT2  {|  cos  2  (n"t  +  p  —  (f>)  +  f  e  cos  (n'^  +  //,  —  TO-)}  sin2  $ 
-zir2{csin^cos6'sin((/)-^Vr)  +  ic2sin2^cos2(</>- JV")}  (6) 

The  dynamical  equations  (§  259) 


sin  0  8(9  VOn 

6  =  —    .        —  f— 
sin  ^  90  VCn 

which  result  must  be  solved  by  continual  approximation.  This  process, 
when  guided  by  the  facts  of  observation  and  limited  to  practical  require- 
ments for  a  period  of  a  century  or  two,  is  very  simple.  For  it  is  known 
that  0  is  very  nearly  constant,  while  <j>  changes  progressively  but  very  slowly. 
Hence  it  is  possible  to  discuss  the  secular  effects,  or  precession,  and  the 
periodic  effects,  or  nutation,  separately. 

263.  The  last  three  lines  in  the  expression  for  U/Cn,  containing  six 
terms,  give  rise  to  periodic  terms  in  6,  <£,  which  can  be  neglected  in  the 
first  instance.  The  secular  changes  come  from  the  terms  in  the  first  line. 
With  sufficient  accuracy  we  may  write 

i  sin  n  =  gt,     i  cos  U  =  g't,     e'  =  e0  +  e:t 

the  quantities  e0,  e1(  g  and  g  being  given  by  the  theory  of  the  Sun's  motion. 
The  corresponding  changes  for  the  Moon  are  negligible  in  effect  or  rather 
are  treated  differently.  Hence  the  equations  for  the  secular  movements  of 
the  Earth's  axis  are 

<j>  =  -{Ka(I-  f  c2  +  fe2)  +  K,  (1  +  |e02)]  cos  0 

—  (Kl  +  K2)         '   (g'  sin  </>  —  g  cos  0)  t  —  3Kle0el .  t  cos  0 
sin  c/ 

6  =      (Kl  +  KZ)  cos  0  (g'  cos  <£  4-  g  sin  <f>)  t. 


300  Precession,  Nutation  and  Time  [CH.  xxn 

When  £  =  0  (1850'0),  Q  is  the  mean  obliquity  of  the  ecliptic  for  that  date 
and  may  be  denoted  by  e0.  Also  $,  being  the  angle  between  the  planes 
OXZ  and  OZz  (§  255),  is  90°  by  the  definition  of  the  axis  OX.  The  periodic 
effects  at  the  time  t  =•  0  are  excluded  from  consideration  here,  but  their 
influence  is  small.  Hence  initially 

<£  =  90°  -  {K2  (1  -  f  c2  +  f  e2)  +  K,  (1  +  f  e02)}  cos  e0 .  t\ 

(  '    sin  f0  '  6°j        I 

6  =  e0  +  I  (K^  +  K2)  cos  e0 .  gt2  ) 

The  length  of  time  during  which  these  expressions  will  be  valid  depends 
on  the  numerical  values  of  the  quantities  involved.  For  a  short  interval 
from  1850'0  (a  century  or  two)  the  preceding  equations  hold  good,  and  may 
be  written 

<f>_  =  90°  —  at.  —  fit?  \ 

(8) 


the  suffix  ra  denoting  mean  values  from  which  periodic  changes  are  excluded. 
Thus  (f>m,  6m  define  the  position  of  the  mean  equator  at  the  time  t  relative  to 
the  fixed  ecliptic  (1850'0),  the  coefficients  a,  /3  and  y  being  now  determined 
by  (7).  The  motion  of  the  mean  equator  on  the  fixed  ecliptic,  measured  by 
90°  —  </>m,  is  called  the  luni-solar  precession  in  longitude.  The  angle  Bm  —  e0 
may  be  called  the  luni-solar  precession  in  obliquity. 

264.  It  has  been  convenient  to  use  a  fixed  set  of  axes  XYZ,  where 
Z  represents  the  pole  of  the  ecliptic  for  1850*0  and  X  the  mean  equinox  for 
the  same  date.  It  is  now  necessary  to  introduce  a  new  set  of  axes  X'Y'Z', 
where  Z'  represents  the  pole  of  the  ecliptic  for  the  epoch  t  and  X'  the 
corresponding  mean  equinox,  i.e.  the  intersection  of  the  mean  equator  and 
ecliptic  at  the  epoch  t.  Let  z  represent  the  N.  pole  of  this  mean  equator, 
its  position  being  defined  by  <f>m,  0m.  The  longitude  of  Z'  in  the  X  YZ  system 
is  ft  -  90°  and  ZZ'  =  i,  where 

i  sin  ft  =  gt  +  ht- 

i  cos  fi  =  g't  +  h'P 

the  terms  of  the  second  order  being  omitted  above  because  they  clearly  give 
rise  to  terms  of  the  third  order  only  in  the  luni-solar  precessions. 

Let  us  consider  the  spherical  triangle  ZZ'z,  of  which  two  sides  are 
ZZ'  =  i  and  Zz=6m.  Since  XZZ'  =  Q,-W  and  XZz  =  $m,  the  angle 
Z'Zz  =  (f)m  —  O  +  90°.  The  side  zZ' ,  which  is  the  mean  obliquity  of  the 
ecliptic  at  t,  will  be  denoted  by  #„/,  and  the  angle  ZzZ',  which  is  called  the 
planetary  precession,  will  be  denoted  by  a.  Hence 

cot  i  sin  0m  =  cos  6m  sin  (O  —  <f>m)  +  cot  a  cos  (H  —  <£m) 


263,  264]  Precession,  Nutation  and  Time 

and  to  the  second  order 

i  cos  (ft  —  <f)m) 


301 


a  = 


cos  i  sin  6m  —  i  sin  (H  —  <£w)  cos  6m 

(g't  +  h't2)  cos  <f>m  +  (gt  +  hP)  sin 


sin  0m  -  f  (gt  +  htf)  cos  </>m  -  (#'£  +  A'£2)  sin  <£TO)  cos  0 


sin  60  +  ^'^  cos  e0 

since  it  is  enough  to  take  dm  =  e0  and  (j>m  =  90°  —  at.     Hence  to  the  required 
order 


at  t2 

a=  -¥  —  4-  -  -  (h  +  ag'  -  qq'  cot  e0) 
sm  e0     sin  e0  v 


Fig.  8. 

Again,  in  the  same  triangle, 

cos  0m'  =  cos  i  cos  0m  +  sin  i  sin  0m  sin  (ft  —  <f>m) 
whence,  to  the  second  order, 

(6m  ~  6m)  sin  \  (0m  +  0mf)  =  -$ia  cos  0m  +  sin  0m  (etgtf  -  g't  - 
To  the  first  order,  therefore, 

0m  —  6m  =  -  g't,     sin  \  (0m  +  0m')  —  sin  e0  +  \g't  cos  e0. 


(9) 


302  Precession,  Nutation  and  Time  [CH.  xxn 

Hence  to  the  second  order 


'      a    -          +  ff'2)  &  cos  €0  +  (g't  +  lit*  -  agt2)  sin  e0 

in          "in  —  =  ~     i     7. 

sin  e0  +  i%g  t  cos  e0 


=  g't  +  h't-  -  agt2  +  %  g"P  cot  e0     ........................  (10) 

The  relations  between  the  various  sets  of  axes  are  shown  in  fig.  8.  The  equator 
X'y  (epoch  t)  cuts  the  fixed  ecliptic  XY  in  x,  where  Xx  =  zZY=QO°  -</>m, 
the  luni-solar  precession,  and  xX'  =  xzX'  =  ZzZ'  =  a,  the  planetary  pre- 
cession. Let  ZX'  cut  X  Y  in  D,  so  that  XD  is  the  negative  mean  longitude 
(1850'0)  of  X',  the  mean  equinox  at  t.  This  arc  is  called  the  general  pre- 
cession and  will  be  denoted  by  90°  —  <£m',  so  that  xD  =  <f>m'  —  <j>,n.  The  angle 
DxX'  =  Zz  =  9m  and  xDX'  is  a  right  angle.  Hence 

tan  (<f>mr  -  <£„,.)  =  tan  a  cos  din 
and  to  the  second  order 


Thus  by  (8)  and  (9)  the  general  precession  may  be  expressed  in  the  form 

90°  -  <£m'  =  Pt  +  P'V 
where 

P  =  «  —  g  cot  e0 

P'  —  /3  -  cot  e0  (h  +  ag'  —  gg'  cot  e0) 
and  by  (8)  and  (10)  the  mean  obliquity  of  the  ecliptic  is 

0m'  =  e0  +  Qt+Q't? 
where 

Q=g' 

Q'  =  y  +  h'  -  ag  +  ^g-  cot  e0. 

265.     To  find  the  periodic  effects,  or  nutation,  it  is  necessary  to  return 
to  §  262  and  write 

</>=<£w  +  <&,   0  =  em  +  ®. 

Now  (f)m  and  6m  have  been  calculated  so  as  to  satisfy  the  secular  terms  which 
arise  in  the  equations  of  motion  from  the  first  line  of  the  expression  (6)  for 
U/Cn.  Hence  the  six  periodic  terms  of  the  last  three  lines  alone  are  now 
relevant,  and  the  dynamical  equations  become 

4>  =  -  KI  {cos  2  (n't  +  fi-<j>)  +  3e'  cos  (n't  +  //  -  BT')J  cos  6 
—  KZ  {cos  2  (n't  +  p  —  <f>)  +  3<?  cos  (n't  +  //,  —  BT)}  cos  6 
-  K2  {c  sin  (0  -  JV)  cos  2(9/sin  ^  +  |c2  cos  2(<f>-N)  cos  ^} 

B  =  {K,  sin  2  (n't  +  //  -  <^>)  +  ^T2  sin  2  (•»"*  +  ^  -  $)}  sin  ^ 
+  7T2  {c  cos  0  cos  (<£  -  N)  -  ^c2  sin  0  sin  2  (</>  -  7^)|. 

The  Moon's  node  makes  a  circuit  of  the  ecliptic  in  18§  years  in  the  retro- 
grade direction,  so  that  it  is  possible  to  write 

N=Nn-N1t. 


264-266]  Precession,  Nutation  and  Time  303 

To  the  first  order  in  t,  which  is  alone  necessary,  6  =  e0  and  </>  =  90°  —  at ;  the 
coefficient  a  can  clearly  be  incorporated  with  n,  n"  and  Nl  before  integration 
in  those  terms  in  which  <£  occurs,  though  the  change  in  n',  n"  is  unimportant. 
Then  on  integration 

f  1  Se 

<&  =  Kl  cos  e0  \-=—.  sin  2  (n't  +  At/) T  sin  (n't  -f  At'  —  *r" 

(2tt  n 

+  K.2  cos  e0  J  77-77  sin  2  (n"i  +  xt) r,  sin  (w"£  +  At  —  t 

[2n  n 

(  c  c2  ) 

-f  KZ  <j=r  sin  (JV0  —  Nj}  cos  2e0/sin  e0  —  -^-  sin  2  (iV0  —  N-f)  cos  e0^ 

0  =  sin  e0  \  ^-.  cos  2  (w'i  +  /n')  4-  ^-77  cos  2  (w"i  +  //,)>• 
(2w  2?i  J 


f  c  c2  ) 


•f     ,  j  •—  cos  60  cos  (  JV0  -  j\fa)  -  ™  sn  e0  cos 

It  is  unnecessary  to  add  integration  constants  because  these  are  incorporated 
in  <f)m  and  0m,  and,  except  as  so  far  explained,  annulled  by  definition  at  the 
initial  epoch  £=0  (1850). 

266.  ©  is  the  nutation  of  the  obliquity  of  the  ecliptic,  and  —  <I>  is  the 
nutation  of  longitude,  </>  and  4>  being  measured  in  the  direction  of  increasing 
longitudes.  The  numerical  quantities  involved  are  of  such  an  order  of 
magnitude  that  a  fair  standard  of  accuracy  has  already  been  obtained  in  the 
formulae.  If  more  precise  results  were  needed,  it  would  be  necessary  (1)  to 
carry  the  expansions  for  the  disturbing  bodies  further,  and  (2)  to  continue 
the  process  of  integration  by  successive  approximation  to  a  higher  stage. 
The  latter  process  would  clearly  introduce  terms  of  the  form  at  sin  (nt  +  a). 
Among  the  terms  of  the  former  origin  those  depending  on  three  times  the 
Sun's  mean  longitude  (n't  +  //)  are  the  most  important,  and  it  may  be  left  as 
an  exercise  to  the  reader  to  determine  them. 

By  far  the  most  important  terms  in  the  nutation  are  those  with  the 
argument  (N0  —  Nj).  The  other  terms  being  omitted,  let 

^=K2ccose0/Nl  ......  .  ....................  (11) 

x  =  [<I>]  sin  e0  =  e/Fsin  (JV0  —  Nj)  cos  2e0/  cos  e0 
y  =  —  [0]       -  -  J''  cos  (N0  -  Nj). 

Since  c/Kis  an  angle  of  a  few  seconds  only,  x  and  y  may  be  considered  as  the 
rectangular  plane  coordinates  of  the  Earth's  pole  relative  to  the  mean  pole, 
x  being  measured  in  the  direction  of  increasing  longitudes  and  y  upwards 
towards  the  pole  of  the  ecliptic.  The  relative  path  of  the  true  pole  is 
therefore  the  small  ellipse 

x2  cos2  e0  +  y'2  cos2  2e0  =  ^  cos2  2e0 


304  Precession,  Nutation  and  Time  [en.  xxn 

described  in  a  period  of  about  18  years.  Since  cos  e0  >  cos  2e0  the  major  axis 
is  directed  towards  the  pole  of  the  ecliptic  and,  since  x  has  the  same  sign 
as  y,  the  sense  of  description  is  such  that  the  relative  longitude  of  the  true 
pole  is  increasing  when  it  lies  between  the  mean  pole  and  the  pole  of  the 
ecliptic,  that  is,  it  is  clockwise  when  viewed  from  a  point  outside  the  celestial 
sphere.  The  centre  of  this  elliptic  motion  is  carried  by  precession  almost 
uniformly  in  the  direction  of  decreasing  longitudes  round  the  pole  of  the 
ecliptic. 

267.  Since  the  manner  of  the  investigation  has  been  controlled  by  the 
actual  magnitude  of  the  various  quantities  involved,  it  is  necessary  to  intro- 
duce numerical  values  if  the  results  are  to  be  properly  understood.  Three 
quantities  are.  based  on  observation,  and  not  derived  from  theory,  namely, 
the  obliquity  e0  at  the  fundamental  epoch  1850'0,  the  precession  constant  P 
and  the  nutation  constant  Jf.  The  values  now  accepted  are 

e0  =  23°  27'  31"-7,     P  =  50"-2453,     </f =  9"'210. 
The  eccentricity  of  the  Earth's  orbit  is  given  by 

e'  =  e0  +  ej  =  O016  7719  -  OOOO  000418 1 
and  the  position  of  the  ecliptic  by 

i  sin  fl  =gt  +  ht2  =  +  0"'05341 1  +  0"'000  01935  P 
i  cos  O  =  g't  +  tit*  =  -  0";46838  t  +  0"'000  00563  F 

the  unit  of  time  being  a  Julian  year  of  365'25  mean  solar  days.  The  Sun's 
period  relative  to  the  equinox  is  the  tropical  year,  and  the  corresponding 
mean  motion  is  therefore 

n'  =  27T  x  365-25/365-2422  =  6'28332. 
The  eccentricity  and  inclination  of  the  Moon's  orbit  are 

e  =  0-05490,     c  =  5°  8'  43"  =  0-089802. 

The  tropical  period  of  the  Moon  is  27'32158  days,  and  hence  the  mean 
motion  in  a  Julian  year  is 

n"  =  83-997  radians. 

The  retrograde  motion  of  the  Moon's  node  has  a  sidereal  period  of  6793  5 
days.  The  corresponding  mean  motion,  corrected  for  precession,  is 

N!  =  0-33757  radians. 
It  is  now  possible  to  derive  the  values  of  Kl  and  K».     In  the  first  place, 

by  (11), 

K*  =  jrNJc  cos  e0  =  37"-74. 
Also 

a  =  P  +  g  cot  e0  =  50"-2453  +  0"'1231  =  50"'3684. 

But,  by  (7)  and  (8), 

a  sec  efl  =  K2  (I  -  f  c2  +  i  e2)  +  K,  (1  +  |e02) 


266-^68]  Precession,  Nutation  and  Time  305 

whence 

54"-91  =  0-992425  7u  +  1-000422  K, 
and  thus 

JTa-lT"^. 

Since  any  error  in  c/F  affects  7T2  directly  and  hence  Kl  equally,  greater  accuracy 
would  be  superfluous.  The  expressions  for  the  luni-solar  precession  (§  263) 
now  become 

90°  -(j>m  =  ctt  +  $V  =  50"-3684  1  -  0"000  1077  t* 

em  =  6fl  +  v?  =  23°  27'  31"-7  +  0"-000  0066  12 
while  the  general  precession  (§  264)  becomes 

90°  -  <£„/  =  Pt  +  P'£2  =  50"-2453  £  -f  0"'000  1107  ? 
and  the  mean  obliquity  of  the  ecliptic 
Om  =  e»+Qt  +  QT- 

=  23°  27'  31"-7  -  0"  -46838  1  -  0""000  0008  1-. 

268.  In  giving  the  numerical  values  of  the  terms  in  the  nutation  (§  265) 
the  notation  is  changed  to  that  employed  in  the  Nautical  Almanac.  The 
results  which  follow  from  substituting  the  above  constants  are  : 

$  =  +  17"'23  sin  S3  -  0"-21  sin  2  £3  +  1"'27  sin  2L 

-  0"-13  sin  (L-ir)  +  0"'21  sin  2  ([  -  0"'07  sin  gl 
(H)  =  +  9"-21  cos  Q  -  0"-09  cos  2  S3  +  0"'55  cos  2Z  +  0"'09  cos  2d 


where  L  is  the  Sun's  mean  longitude  (n't  +  fj!),  TT  is  the  longitude  of  the 
Sun's  perigee  (-or'),  ((  is  the  Moon's  mean  longitude  (n't  +  p),  gl  is  the  Moon's 
mean  anomaly  (ri't  +  p  —  VT),  and  S3  is  the  longitude  of  the  Moon's  ascending 
node  (N0  —  Nj).  In  the  Nautical  Almanac  the  nutation  of  the  obliquity  of 
the  ecliptic  (©)  is  called  A&>,  and  the  nutation  of  longitude  (—  <3>)  is  called 
AX.  Comparison  shows  that  no  term  with  coefficient  exceeding  0"'05  has 
been  omitted  here. 

Two  important  astronomical  constants  are  involved  implicitly  in  the 
constants  of  nutation  and  precession,  namely  the  mass  of  the  Moon  and  the 
ratio  (C—A)/C,  which  has  been  called  the  mechanical  elliptic!  ty  of  the 
Earth.  For  the  equations  (5)  may  be  written 


__ 
1+/~#1'«"2'       '  C     ~3'   »'» 

the  mass  of  the  Earth,  E=  1/333432,  being  negligible.  Here  K^  and  K2, 
expressed  above  in  seconds  of  arc,  are  angular  motions  in  a  Julian  year, 
and  n,  n'  and  n"  are  sidereal  mean  motions  in  the  same  unit  of  time.  With 
sufficient  accuracy  the  above  values  of  n'  and  n"  may  be  used,  and  for  n  the 
value  2?r  x  366^.  Hence 

//(I  +/)  =  0-012102,    /=  1  /81-6 
p.  D.  A.  20 


306  Precession,  Nutation  and  Time  [CH.  xxn 

for/",  the  ratio  of  the  mass  of  the  Moon  to  the  mass  of  the  Earth,  and 

C-A       JL 
C     *  304-2 

for  the  mechanical  ellipticity  of  the  Earth.  The  mass  of  the  Moon  is  also 
obtained  as  a  by-product  from  the  observations  of  a  minor  planet  in  a  refined 
determination  of  the  solar  parallax.  The  value  of  f  found  by  Hinks  in  this 
way  was  I/ 81 '53. 

269.  The  practical  application  of  the  results  obtained  for  precession  and 
nutation  belongs  to  the  domain  of  Spherical  Astronomy  and  will  not  be 
pursued  in  detail  here.  Nutation  is  so  small  that  its  effects  can  be,  and 
are,  treated  independently  of  those  due  to  precession.  Of  the  latter  some- 
thing more  may  be  said  in  order  to  define  the  two  quantities  employed  in 
calculating  the  effects  of  precession  in  right  ascension  and  declination. 

Let  a,  8  be  the  R.A.  and  declination  of  a  star  at  the  epoch  t.  These  refer 
to  the  system  of  axes  X'y'z  (fig.  8),  which  differs  by  a  simple  rotation 
through  the  angle  a  about  z  from  the  system  xyz.  Hence  the  coordinates 
of  the  star  in  the  latter  system  are 

x  =  cos  8  cos  (a  +  a),     y  —  cos  8  sin  (a  +  a),     z  =  sin  8 
whence,  by  differentiation  with  respect  to  t,  it  easily  follows  that 

a  4-  a  —  (xy  —  y#)/cos2  8 
8  =  z  I  cos  8. 

Now  the  relations  between  the  systems  xyz  and  XYZ  are  expressed  by  the 

scheme : 

X  Y  Z 

x  sin  (f>  —  cos  (f)  0 

y         cos  0  cos  (f>  cos  0  sin  <£         —  sin  0 

z         sin  0  cos  (j>  sin  0  sin  <£  cos  0. 

Here  XYZ  are  constant,  and  differentiation  of  the  linear  formulae  for  xyz, 
when  XYZ  are  finally  expressed  in  terms  of  x,  y,  z,  gives 

x  =  (y  cos  0  +  z  sin  0)  <£ 
y  =  —  x  cos  0 .  (j>  —  z0 
z  =  -  x  sin  0  .  (j>  +  yd. 
Hence,  when  x,  y,  z  are  expressed  in  terms  of  a,  8, 

d  +  d  =  -  cos  0 .  £  -  tan  8  sin  (a  +  a)  sin  0 .  <j>  -  tan  8  cos  (a  +  a)  .  0 
8  =  -  cos  (a  4-  a)  sin  0 .  <j>  +  sin  (a  +  a)  0. 

These  differential  expressions  are  required  to  the  first  order  in  t,  and  a0 
being  of  the  second  order  may  be  rejected  at  once.  Hence  (the  symbol  n 
being  used  here  in  a  new  sense) 


268-270]  Precession,  Nutation  and  Time  307 

d  =  m  +  n  sin  a  tan  &  —  p  cos  a  tan  B 

8  —  n  cos  a  +p  sin  a 
where 

in  =  —  d  —  cos  6 .  <p,     n  =  —  sin  6  .  <£,    p  =  a  sin  0 .  <j>  +  $ 

and  #  may  be  replaced  by  e0.  With  the  numerical  values  given  in  §  267,  (9) 
gives 

a  =  +  0"-1342 1  -  0"-000  2380  t2 

a  =  +  0"-1342    -  0"-000  4760  t 
and  from  the  luni-solar  precessions 

0  =  -  50"-3684  +  0"-000  2154 1 

e=  +  o"-ooo  0132 1. 

Hence 

m  =  +  46"'07ll  +  0"-000  2784 1 

n  =  +  20"'0511  --  0"'000  0857  t 

while  p  =  +  0"'000  0002  and  is  altogether  negligible.  Thus  m  and  n  are  the 
important  quantities  known  as  the  annual  precessions  in  R.A.  and  declination. 
The  total  precession  in  R.A.  from  1850  for  a  point  on  the  equator  is 

mdt  =  mj  +  m2t2  =  46"'07ll  t  +  0"'000  1392  V. 

The  expressions  found  for  d,  S  are  the  coefficients  of  the  first  power  of  the 
time  and  these  terms  suffice  for  short  intervals  only.  The  further  develop- 
ment of  formulae  for  the  transformation  of  coordinates  from  one  epoch  to 
another  according  to  the  methods  of  astronomical  practice  must  be  sought  in 
such  works  as  Newcomb's  Compendium  of  Spherical  Astronomy. 

270.     It   is  now  possible  to  consider   in  some  detail  the  astronomical 
measure  of  time.     The  third  equation  of  (1)  is 

a)3  =  ijr  +  <j)  cos  9. 

Here  <w3  is  the  angular  velocity  of  the  Earth  about  its  axis  of  figure  and  is 
a  constant  previously  denoted  by  n.  As  this  symbol  has  been  used  with 
another  meaning  in  §  269  it  will  now  be  replaced  by  eo.  The  angle  ty  is 
the  angle  between,  a  meridian  plane  (Ozx)  fixed  in  the  Earth  and  rotating 
with  it  and  the  plane  (OZz)  passing  through  the  pole  of  the  fixed  ecliptic. 
For  the  fixed  meridian  we  adopt  the  meridian  of  Greenwich.  The  rotation 
•^r  refers  therefore  to  the  Greenwich  meridian  relative  to  zx  in  fig.  8,  and 
T  =  ^r  —  a  will  measure  the  same  rotation  relative  to  zx.  But  the  angle 
between  the  Greenwich  meridian  and  zx,  x  being  the  equinoctial  point  at 
the  time  t,  is  the  hour-angle  of  the  First  Point  of  Aries,  i.e.  the  sidereal  time 
at  Greenwich.  Thus,  r  being  Greenwich  sidereal  time, 

f  =  ^jr  —  d  =  w  —  a  —  (j>  cos  6. 

20—2 


308  Precession,  Nutation  and  Time  [CH.  xxn 

It  is  the  true  equinox  which  is  now  involved,  affected  both  by  precession  and 
nutation,  so  that 

Hence 

f  =  &)  —  a  —  (f)m  cos  6m  —  <t>  cos  6  +  (j>m  ®  sin  6m 

=  w  -f-  m  —  <J>  cos  6  —  n® 
=  co  +  m  —  4>  cos  e0 

with  sufficient  accuracy,  for  n®  can  be  neglected  since  ®  -is  small  and  n 
is  about  10~4,  and  d>  being  small  cos  6  may  be  replaced  by  cos  e0.  Hence 
integration  gives  for  Greenwich  sidereal  time 

T  =  T0  +  (at  +  mj  +  m.2t2  —  <I>  cos  e0    (12) 

where  t  is  measured  in  Julian  years  of  365'25  mean  days  and  reckoned  from 
1850  Jan.  0,  Gr.  mean  noon.  The  quantity  t  is  an  equi- crescent  variable  in 
the  sense  required  by  the  dynamical  laws  which  have  been  used ;  its  origin 
and  unit  are  for  the  moment  of  importance  only  so  far  as  they  condition  the 
numerical  values  of  the  coefficients.  On  the  other  hand  the  sidereal  time  T 
is  not  uniform,  being  affected  by  secular  and  periodic  terms.  Hence  T  is 
merely  an  intermediate  standard  of  time.  But  this  in  no  way  affects  its 
practical  utility.  By  far  the  largest  term  in  <J>  cos  e0  is 

15"'803  sin  &  =  18'054  sin  £ 

of  which  the  period  is  nearly  19  years,  and  m.2  is  very  small.  The  irregularities 
in  T  are  therefore  very  small  and  gradual,  and  far  less  than  the  natural 
irregularities  in  the  rate  of  the  most  perfect  sidereal  clock.  Since  this 
instrument  shows  the  hour-angle  of  the  First  Point  of  Aries,  it  also  shows 
the  right  ascension  of  stars  on  the  meridian,  and  this  principle  serves  both 
to  determine  the  error  of  the  clock  and  to  measure  the  apparent  positions  of 
the  stars. 

271.  In  the  next  place  a  mean  Sun  is  defined  which  moves  in  the  plane 
of  the  equator  with  the  uniform  sidereal  mean  motion  /u,.  Its  R.A.  at  time  t, 
reckoned  from  the  true  equinox,  is  therefore 

A  —  A  0  +  /j,t  +  nijt  +  m2£2  —  <&  cos  e0 
and  its  hour-angle 

T  =  T  —  A  =  T0  —  A0  +  (w  —  ft)  t 

is  the  measure  of  Greenwich  mean  time.  The  constants  occurring  in  A  are 
adjusted  as  far  as  possible  to  secure  identity  with  the  mean  longitude  of  the 
actual  Sun  affected  by  aberration.  This  may  be  written  in  the  form 

7"    /-\        I     \    +     l     \    +2\  Z/.     I     /  P/     I       f>'tZ\ 

Li  —  1  A,n  ~T  Art*  T  A.2C   )  —  hi  -f-  ( JLt/  -\-  JL     v   I 


270,  271  ]  Precession,  Nutation  and  Time  300 

where  A0  is  the  true  mean  longitude  of  the  Sun  when  t  =  0,  Xx  is  the  sidereal 
mean  motion,  and  2X2  is  the  secular  acceleration  which  arises  indirectly  from 
the  perturbations  of  the  other  elements  of  the  Earth's  orbit  ;  k  =  20"'47  is 
the  constant  of  aberration;  and  (Pt  +  P't2)  is  the  general  precession  in 
longitude.  The  adjustment  of  the  constants  in  A  and  L  gives 


and  leaves  outstanding  between  L  and  A  the  secular  discrepancy  (L2  —  ?u2)  t2 
which  would  lead  ultimately  to  a  departure  of  the  actual  Sun,  apart  from 
periodic  effects,  from  the  meridian  at  mean  noon.  This  quantity  is  small 
and  far  from  certain  in  amount,  and  will  have  no  practical  effect  for  many 
centuries  to  come.  Now  at  1850  Jan.  0,  Greenwich  mean  noon, 

T=t=0,     r0=A0  =  L0 
and  the  effect  of  adding  one  mean  day  to  T  or  t  is 

24h  =  360°  -  (w  -  /*)/365'25 
whence 

6)/365'25  =  24h  +  (L,  -  w^/  365-25 

(a)  +  wij)/  365-25  =  24h  +  A/  365-25. 
Now,  according  to  Newcomb, 

L,  =  279°  47'  58"-2  =  18h  39m  118'88 
L,=  1296027"-6674  =  8640P-84449 
L2  =  +  0"'000  1089  =  +  08-000  00726 

while  in  the  latter  unit  (1s  =  15") 

mi  =  +3s-07!41,     wia  =  +  08-000  00928 
so  that 

A/365-25  =  2368-55533,     (L,  -  7?i1)/365'25  =  2368'54692. 

Hence  in  numbers  the  equation  (12)  for  Gr.  sidereal  time  becomes 
T  =  18h  39m  118'88  +  (24h  3ra  568'55533)  D  +  08'000  00928  12  -  $  cos  e0 

where  D  =  365'25  t  is  the  number  of  days  reckoned  from  1850  Jan.  0.  When 
D  is  given  an  integral  value  this  expression  gives  the  sidereal  time  at  Gr. 
mean  noon  and  its  value  (less  a  multiple  of  24h)  is  tabulated  for  every  day 
in  the  Nautical  Almanac.  When  the  nutational  term  is  omitted, 

AT  =  (24h  3m  568'55533  +  08'000  00005  1)  AD. 
The  secular  term  is  also  negligible,  and  hence 

1  mean  day   _  86636-55_5  _ 
1  sidereal  day  ~     86400s 


310  Precession,  Nutation  and  Time  [CH.  xxn 

Another  period  which  differs  little  from  the  sidereal  day,  but  must  not  be 
confounded  with  it,  is  the  period  of  the  Earth's  rotation  on  its  axis,  measured 
by  o>.  Its  ratio  to  the  mean  sidereal  day  is 

&>  +  m,     86636-555 

=  HTvqr-Kk'7  =  1'000  00°  °97- 
&>          86636547 

272.  A  catalogue  of  astronomical  positions  gives  mean  places  freed  from 
nutation  and  reduced  to  the  equinox  of  a  common  epoch.  Such  an  epoch  is 
always  the  beginning  of  a  tropical  year  and  this  expression  must  be  defined. 
It  is  the  moment  when  the  mean  longitude  of  the  Sun  as  above  described, 

L  =  L0  +  Lj  +  L.tf 
is  280°  =  18h  40m.     It  follows  that  the  length  of  a  tropical  year  is 

24h 

•f -ETf~.  •  365'25  mean  days 

A  +  zLJ  J 

\ 

365-25 


1-000  021  3483  +  O'OOO  000  000  168 1 
=  365-242  20272  -  O'OOO  000  0614 1 

or  365-242200  mean  solar  days  at  the  epoch  1900.  For  the  present  the 
secular  change  is  unimportant.  Once  the  beginning  of  the  tropical  year 
is  fixed  in  a  particular  calendar  year,  its  beginning  in  any  other  year 
may  be  found  by  adding  so  many  tropical  years.  But  the  details  will  be 
better  illustrated  by  a  direct  example  from  the  year  1900.  When  £  =  50, 
L'=  18h40m44"123.  Now  50  Julian  years  exceed  50  years  of  365  days  by 
12|  days,  whereas  the  calendar  inserts  12  leap  days  between  1850  and  1900. 
Hence  this  is  the  mean  longitude  for  1900  Jan.  0'5.  The  mean  longitude 
for  1900  Jan.  0  (Gr.  mean  noon)  is  therefore  L'  -  IA/365'25  =  18h  38ra  45S>845 
and  must  be  increased  by  748'155  at  the  daily  rate  2368>555  in  order  to 
become  18h  40m.  This  requires  0'3135  mean  days,  and  the  beginning  of 
the  tropical  year  in  1900  is  therefore  Jan.  0'3135,  the  fraction  of  a  mean  day 
being  reckoned  from  Greenwich  mean  noon.  This  epoch  is  recorded  briefly 
as  1900*0.  It  is  to  the  mean  equinox  of  this  date  that  the  observations  of 
the  year  are  reduced  in  the  first  instance. 

273.  Such  in  outline  are  the  main  features  in  the  astronomical  methods 
of  reckoning  time.  They  involve  certain  constants  which,  being  based  on 
the  comparison  of  theory  with  observations,  are  capable  of  improvement. 
But  there  is  no  absolute  standard  of  time.  Ultimately  no  doubt  the  con- 
tinued comparison  of  theory  with  observation  according  to  such  a  system  of 
time  as  that  described  above  will  bring  to  light  discrepancies  in  the  motions 
of  the  heavenly  bodies  of  a  kind  which  cannot  be  attributed  to  errors  of 


271-273]  Precession,  Nutation  and  Time  311 

observation.  Then  the  question  will  arise  whether  these  discrepancies  can 
be  removed  by  a  mere  adjustment  of  an  accepted  system  of  constants  in- 
volved in  the  measure  of  time  or  whether  the  fault  lies  in  the  theory.  This 
is  the  ordinary  experience  of  practical  astronomy.  It  may,  however,  prove 
that  what  have  been  regarded  as  constants  are  not  really  constant  at  all. 
Thus  <w,  the  rate  of  rotation  of  the  Earth  on  its  axis,  may  vary  owing  to  such 
causes  as  the  secular  cooling  of  the  Earth  and  the  effect  of  tidal  friction. 
There  is,  indeed,  reason  to  think  that  this  is  so.  But  ultimately  it  is  only 
possible  to  adopt  such  a  system  of  measuring  time  as  will  reconcile  all 
celestial  phenomena  as  far  as  may  be  with  the  simplest  possible  body  of 
laws.  In  the  meantime  to  deal  with  discrepancies  as  they  arise  is  among 
the  most  critical  problems  of  technical  astronomy. 


CHAPTER   XXIII 

LIBRATION    OF    THE    MOON 

274.  The  form  of  solution  found  suitable  in  discussing  the  rotation  of 
the    Earth  depends   on  special  circumstances  and  is  by  no  means  general. 
The  Moon's  rotation  similarly  presents  quite  special  features  which  require 
very  different  treatment.     This  movement  is  governed  to  a  high  degree  of 
approximation  by  Cassini's  laws  : 

(1)  The  Moon   rotates  uniformly  about   an   axis  which    is  fixed  with 
respect  to  the  Moon  itself.     The  period  of  this  rotation  is  identical  with  the 
sidereal  period  of  the  Moon  in  its  orbit,  namely  27'321661  days. 

(2)  The  pole  of  the  lunar  rotation  z  makes  a  constant  angle  (1°  35') 
with  the  pole  of  the  ecliptic  Z,  which  may  here  be  regarded  as  a  fixed  point 
on  the  celestial  sphere. 

(3)  In  consequence  of  the  nearly  uniform  regression  of  the  lunar  node 
on  the  plane  of  the  ecliptic  and  the  nearly  constant  inclination  of  the  lunar 
orbit  (5°  9'),  the  pole  of  the  Moon's  orbit  P  is  known  to  describe  a  small 
circle  about  Z  in  a  period  of  18|  years.    The  arc  of  a  great  circle  zP  contains 
also  the  pole  Z.     In  other  words,  the  planes  of  the  lunar  orbit  and  the  lunar 
equator  intersect  on  the  ecliptic,  the  latter  plane  being  intermediate  between 
the  two  former. 

These  laws  were  discovered  by  observation  and  they  are  so  exact  that 
later  work  with  more  refined  instruments  has  failed  hitherto  to  determine 
any  divergences  from  them  with  a  satisfactory  degree  of  certainty.  They 
define  as  it  were  a  steady  state  of  motion,  and  it  is  necessary  to  inquire 
under  what  conditions  such  a  state  is  possible,  and  to  what  oscillations  it  is 
subject  according  to  theory. 

275.  The  first  of  the  above  laws  corresponds  with  the  well-known  fact 
that  the  Moon  always  presents  the  same  face  to  the  Earth,  or  more  truly 
that  a  large  fraction  of  its  surface  (nearly  f )  is  always  concealed  from  obser- 
vation.    In  order  that  exactly  the  same  face  should  be  seen  at  all  times 
three  further  conditions  would  be  necessary  and  the  failure  of  these  conditions 
gives  rise  to  three  distinct  components  of  what  is  called  the  apparent  or 


274-276]  Libration  of  the  Moon  313 

optical   libration   of  the    Moon.     These   conditions   and    the   corresponding 
effects  of  their  departure  from  the  facts  are  : 

(1)  The  motion  of  the  Moon  in  its  orbit  about  the    Earth  must  be 
uniform.    But  owing  to  the  equation  of  the  centre  and  periodic  perturbations 
the  actual  place  of  the  Moon  may  differ  from  its  mean  place  by  as  much  as 
8°.     Hence  an  oscillation  in  the  central  meridian,  which  is  known  as  the 
libration  in  longitude. 

(2)  The  axis  of  the  Moon  must  be  normal  to  the  plane  of  its  orbit. 
Actually  the  angle  which  it  makes  with  the  normal  to  the  orbit  is 

1°  35'  +  5°  9'  =  6°  44'. 
The  monthly  effect  of  this  is  called  the  libration  in  latitude. 

(3)  The  point  of  observation  must  be  the  centre  of  the  Earth.     Owing 
to  the  position  of  the  observer  on  the  Earth's  surface,  which  varies  with  the 
rotation  of  the  Earth,  there  is  a  parallactic  effect  which  is  called  the  diurnal 
libration. 

These  three  effects  which  together  constitute  the  optical  libration  of  the 
Moon  are  purely  geometrical  consequences  of  the  known  conditions,  and 
entirely  independent  of  the  dynamical  libration  which  is  now  to  be  examined. 

276.  When  the  rotation  of  the  Moon  is  in  question  the  action  of  the 
Earth  as  a  disturbing  body  is  clearly  preponderant  and  the  action  of  the 
Sun  is  neglected.  Let  0  be  the  centre  of  gravity  of  the  Moon,  OXYZ  a  set 
of  ecliptic  axes,  fixed  in  space,  and  Oxyz  a  set  fixed  in  the  rotating  body  and 
coinciding  with  the  principal  axes  of  the  Moon,  the  corresponding  moments 
of  inertia  being  A,  B,  C.  Now  since  the  axis  of  rotation  is  nearly  or  quite 
fixed  in  the  body  it  must  practically  coincide  with  a  principal  axis  ;  for  a 
permanent  axis  in  any  other  position  would  require  a  constraint  which  is 
obviously  absent  in  this  case.  This  principal  axis  will  be  identified  with  Oz. 
As  in  §  255  the  two  sets  of  axes  are  connected  by  the  angles  6,  <£  and  T/T,  and 
6  =  ZOz  being  always  of  the  order  10>6,  its  square  may  be  neglected.  The 
relations  between  the  coordinates  are  then  given  by  the  scheme  : 

X  Y  Z 

X  COS  ((£  +  \|r)  sin  ((j)  -f  ty)  —0  COS  *fr 

y     —  sin  (</>  +  -v/r)         cos  ($  +  >/r)  8  sin  i/r 

z  9  cos  (f>  6  sin  <£  1 

and  Euler's  geometrical  equations  become 

&>!  =  6  sin  \Jr  —    0  cos  ir 


o>2  =     cos   r  +  (>    sn 


314  Libration  of  the  Moon  [CH.  xxm 

* 

The  dynamical  equations  are  again  of  the  form 

A6*!  —  (B—C)  o)2o)3  =  L 
Ba>2  -  (C  -  A)  0)3^  =  M 

Cw,  -  (A  -  B)  Wla),=  N 
where  (§  257) 

l  =  3Gm(C-B)yz/r*,     M=  3Gm  (A  -  C)  xz\r\     N=  3Gm  (B  -  A}  xy/r* 

m  being  the  mass  of  the  Earth,  (oc,  y,  z)  its  coordinates  and  r  its  distance 
from  the  Moon.  Let  (X,  Y,  Z)  be  the  ecliptic  coordinates  of  the  Earth 
relative  to  the  Moon.  The  inclination  of  the  Moon's  orbit,  c  =  5°  9',  is  so 
small  that  c2  will  be  neglected.  Then  (cf.  §  65) 

-  X  =  r  cos  (H  +  «  +  w),      -  Y  =  r  sin  (II  +  <o  +  w),     —  Z  =  re  sin  (w  +  w) 

where  H  is  the  longitude  of  the  Moon's  node,  (£1  +  &>)  the  longitude  of  the 
Moon's  perigee,  and  w  the  Moon's  true  anomaly.  But 

X=  fl  +  w  +  w 

is  the  longitude  of  the  Moon  in  its  orbit.  Hence,  by  the  above  relations 
between  the  two  sets  of  coordinates, 

—  x  =  r  cos  (X,  —  <£  —  -v/r),     —  y  =  r  sin  (X  —  $  —  ^r) 

—  z=r6  cos  (A.  —  </>)  +  re  sin  (A,  —  O) 
the  product  c6  being  neglected  in  x  and  y.     Let 

C-B  =  Aa,     A-C=*B/3,     B  -  A  =  Cy. 
Then  the  dynamical  equations  of  motion  become 
&>!  +  a«2<»3  =  36rwar-3  sin  (X  —  0  —  T/T)  {#  cos  (A,  —  0)  +  c  sin  (A,  —  fl)h 


3  sin  2  (X  —  <£  —  i/r) 

As  the  figure  of  the  Moon  is  to  all  appearance  sensibly  spherical,  a,  /3  and  7 
must  be  fairly  small  quantities.  And  since,  further,  the  instantaneous  axis 
is  nearly  fixed  in  the  body  and  very  close  to  the  axis  of  z,  o^  and  o>2  must  be 
very  small  in  comparison  with  &>3. 

277.     It  follows  that  in  the  last  equation  the  term  yw1  <w2  can  be  neglected. 
Hence  this  equation  becomes,  in  view  of  the  third  geometrical  equation, 

<£  +  £-f  £77177--*  sin  2(X-<£-<f)    ..................  (2) 

The  Moon's  mean  longitude  is  n't  +  e,  where  ri  is  the  Moon's  mean  motion 
and  e  is  a  constant.  The  Earth's  mean  longitude,  as  seen  from  the  Moon,  is 
therefore  IT  +  n't  -f  e.  But  according  to  Cassini's  first  law, 

®3  =  0  +  ^  =  ri 
or 

<£  +  -vjr  =  n't  +  const. 


276-278 J  Lib-ration  of  the  Moon  315 

the  constant  depending  on  the  choice  of  a  fixed  meridian  on  the  Moon's 
surface.  Let  it  be  so  chosen  that  the  latter  expression  is  equal  to  the 
Earth's  mean  longitude.  The  corresponding  meridian  is  called  the  first 
lunar  meridian.  In  order  now  to  allow  for  a  possible  inequality  in  the 
Moon's  rotation  an  angle  %  is  introduced  such  that 

<£  +  -f  +  x  =  Tr+n't  +  e ,....(3) 

This  angle  represents  an  oscillation  in  the  position  of  the  first  meridian. 
According  to  Cassini's  laws  %  =  0  and  observation  proves  that  ^  is  certainly 
very  small.  The  equation  (2)  now  becomes 

X  =  —  |  Gmyr~s  sin  2  (^  +  X  —  n't  —  e)   (4) 

It  is  clear  that  the  conditions  of  stability  are  only  complicated  by  the 
inequalities  in  the  motion  of  the  Moon.  Therefore  we  substitute  for  the 
moment  a  uniform  circular  orbit  with  mean  distance  a,  so  that  \  =  n't  +  e, 
r  =  a'  and 

%  =  —  |  Gmya'~s  sin  2^ 

=  -%n*y(l+f)-*BSai2x     (5) 

where /is  the  ratio  of  the  mass  of  the  Moon  to  the  mass  of  the  Earth ;  since 
by  Kepler's  third  law 

Gm(l+f)  =  n'*a'*     (6) 

But  the  equation  of  motion  of  a  simple  pendulum  of  length  I  and  inclined  to 
the  vertical  at  an  angle  6  is 

0  =  -  gl~l  sin  0 

which  can  be  identified  with  (5)  by  taking  %=  \Q  and  3ri'2y(l  +f}~l  =  gl-*. 
Both  equations  can  of  course  be  solved  generally  in  elliptic  integrals.  But 
it  is  enough  to  notice  the  physical  fact  that  the  pendulum  is  capable  of 
small  vibrations  provided  6  is  small  initially  and  g  is  positive.  Similarly  ^ 
if  initially  small  will  remain  small  provided  7  is  positive,  i.e.  B  >  A.  Now,  if 
the  inclination  "of  the  lunar  equator  to  the  lunar  orbit  be  neglected,  (0  +  -^) 
measures  the  displacement  of  the  axis  of  x  from  the  equinox  from  which  the 
longitudes  are  reckoned.  Under  these  simplified  conditions  the  first  meridian 
contains  the  axis  of  x  and  always  coincides  with  the  central  meridian  of  the 
apparent  disc.  The  axis  of  x  is  therefore  directed  approximately  towards 
the  Earth  and  this  defines  the  axis  about  which  the  moment  A  is  less  than 
the  moment  B.  This  is  the  first  condition  of  stability.  It  is  also  to  be 
inferred  that  A  ^  B.  For  if  A  =  B,  %  =  0  and  a  small  disturbance  would 
introduce  a  secular  term  in  ^  which  observation  shows  to  be  absent. 

278.     If  7'  =  7(1  +f)~l  the  more  general  equation  (4)  for  ^  becomes 
%  =  —  f  ?i'V  (a'/r)3  sin  2  (^  +  X  —  n't  —  e). 

Now  (A,  —  n't  —  e)  is  of  the  order  of  the  eccentricity  of  the  lunar  orbit 
(•055).  x  is  still  smaller  and  a  /r  differs  from  1  also  by  a  quantity  of  the 


316  Libration  of  the  Moon  [OH.  xxm 

order  of  the  eccentricity.  Hence  if  the  square  of  the  eccentricity  be 
neglected, 

X  =  -  3n'y  (x  +  \  -  n't  -  e) 
or 

X  +  3wV%  =  -  3n'V  2#  sin  (ht  +  hf) 

where  the  terms  under  S  represent  the  equation  of  the  centre  and  periodic 
inequalities  of  the  lunar  motion.  This  is  the  ordinary  equation  for  forced 
vibrations  and  the  solution  may  be  written  in  the  form  x  —  X*  +  X*  wnere  %i 
is  a  particular  solution,  corresponding  to  the  forced  vibrations,  and  ^2  is  the 
complementary  function,  corresponding  to  an  arbitrary  free  vibration.  It  is 
easily  verified  that 


and 

Xz  =  K  sin  [n't  >J(3y)  +  k'] 

where  K,  k'  are  arbitrary.  Terms  in  Xi  can  onty  become  sensible  by  reason 
of  H  large  or  h  small,  and  the  most  promising  terms  in  the  lunar  theory  are 
consequently  the  equation  of  the  centre  (or  principal  elliptic  term)  : 

ht  +h'=glt     H=  +  22639"-!,     h  =  47033"'97 
and  the  annual  equation  : 

ht  +  h'  =  ©,     H=-  668"-9,     h  =  3548"-!  6 

where  gl  is  the  Moon's  mean  anomaly,  ©  is  the  Sun's  mean  anomaly,  and 
the  unit  of  time  is  the  mean  solar  day,  so  that  n'  —  47435"-03.  The  corre- 
sponding terms  in  x\  are 

377'  11'15 

*  =  6=3277^7  '  7  Sm  *  ~  6=661865^7  '  7  8m  G 

It  is  easily  seen  that,  7'  being  certainly  very  small,  it  is  the  second  of  these 
terms  which  is  the  larger.  But  the  determination  of  its  coefficient  from 
observation  has  not  yet  been  made  with  satisfactory  certainty.  Since  the 
Earth's  distance  is  about  220  times  the  Moon's  radius  a  geocentric  angle 
of  1"  is  the  equivalent  of  4'  in  selenographic  arc  near  the  centre  of  the  lunar 
disc.  As  the  quantities  to  be  looked  for  are  likely  to  be  of  this  order,  or 
rather  still  less,  and  the  observations  are  very  difficult,  positive  results  must 
be  awaited  from  the  study  of  the  large-scale  photographs  of  the  Moon  which 
are  now  available.  According  to  Franz,  using  the  heliometer  observations  of 
Schliiter,  the  coefficient  of  sin  0  is  about  2',  giving  7  of  the  order  0'0003, 
and  the  arbitrary  libration  K,  which  should  have  a  period  of  rather  more 
than  2  years,  is  practically  negligible. 

279.     Since,  by  (3),  &>3  -I-  x  =  n'  where  x  may  now  be  supposed  very  small, 
the  first  two  dynamical  equations  may  be  written 

=L/A\ 


278,  279]  Libration  of  the  Moon  317 

Now  let 

£  =  6  cos  -fy,     i)  =  6  sin  -v/r 
so  that 

%  =  0  cos  ty  +  $0  sin  -fy  —  (<j>  +  i^)  0  sin  \|r  =  to2  —  ta^l 

r  .........  (9) 

77  =  6  sin  i|r  —  $6  cos  ^  +  ($  +  ty)  6  cos  i|r  =  &>j  +  <wsf  J 

Again  o>3  may  be  replaced  by  n',  being  multiplied  by  £  and  77  which  are 
small.     Hence  (8)  become 

77  -  (1  -  a)  w'i  +  cm'2??    =  X  /  A 


Expressions  for  L/A,  MjB  have  been  given  in  (1),  and  if  /=  1/81  be 
neglected  in  (6)  these  are 

L!  A  =  Saw'2  (a'/r)3  sin  (\-<f>~  -f)  [d  cos  (X  -  <f>)  +  c  sin  (X  -  II)} 
.¥  /£  =  3/3w'2  (a'/r)2  cos  (\  -  0  -  -f  )  (0  cos  (X  -  0)  4-  c  sin  (X  -  H)} 

and  as  they  are  already  of  the  order  6  or  c  multiplied  by  a  or  ft,  the  other 
quantities  involved  are  only  required  to  the  first  order  in  e,  the  eccentricity 
of  the  orbit.  Now  gl  being  the  mean  anomaly,  by  Ch.  IV  (9)  and  (30)  —  or 
in  a  more  simple  way  — 

a  IT  =  1  +  ecosf/j,     w  —  g^  =  2esin^! 
where 

g1  =  n't  +  €  —  •&-,     w  =  \  —  CT 

w  being  the  true  anomaly  and  •57  the  longitude  of  perigee.  Also  ^  is  in- 
significant here,  so  that  by  (3) 

(f>  +  \lr  =  7r  +  n't-}-e  =  gl  +  'GT  +  7r  .....................  (10) 

Hence 

X  —  0  —  ty  =  lu  —  gl  —  TT  =  2e  sin  gl  —  TT 

sin  (X  —  <f>  —  >/r)  =  —  2e  sin  gl  ,     cos  (X  —  (f>  —  -^)  =  —  1 


(a'/r)3  cos  (X  -  <£  —  -v|r)  =  —  1  —  3e  cos  ^  J 
Again, 

cos  (X  —  (£)  =  —  cos  (i/r  +  2e  sin  ^rj  =  —  cos  -fr  •+  2e  sin  ^  sin  ty 

6  cos(X  —  <^>)  =  —  6  cos  i/r  +  e0cos(gl  —  ^)  —  e6cos(gl  +  i/r)  ...(12) 

and  finally 

X  —  £1  =  w  +  •&  —  O  =  ^1+/nr  —  fl  +  2e  sin  </j 

sin  (X  -  fl)  =  sin  (  ^  +  ta  —  fl)  4-  2e  sin  ^  cos  (^  +  w  -  O) 

c  sin  (X  —  H)  =  c  sin  (g1  +  TX  —  H) 

+  06801(2^!+  OT  —  H)  —  cesin(t3-  —  O)  ............  (13) 

It  is  now  necessary  to  introduce  (11),  (12)  and  (13)  into  L/A,  MjB,  to 
reject  terms  of  the  third  order  in  e,  c  and  6,  arid  to  resolve  the  products 


318  Libratiou  of  the  Moon  [CH.  xxni 

of  circular  functions  which  occur  into  single  functions.     The  result  of  this 
simple  reduction  gives 

Lj  A  =  3cm'2  {  ed  sin  (c/i  +  -^)  +eO  sin  (gl  —  ty)  —ec  cos  (-or  -  ft) 


M/B  =  3/8w'*  {f  e6  cos  (^  +  ^r)  +  i  ed  cos  (gl  -  ^r)  -  \ec  sin  (-57  -  ft) 
—  fee  sin  (2^  +  OT  —  ft)  —  c  sin  (g1  +  ^  —  Cl)  +  6  cos  -^J 

The  last  term  in  MjB  is  3/8w'2£,  which  may  be  transferred  immediately  to 
the  other  side  of  the  corresponding  dynamical  equation.  This  leaves  one 
term  only  of  the  first  order  in  M/B  :  the  remaining  terms  in  L/  A  and  M/B 
are  entirely  of  the  second  order. 

280.     Let  the  actual  dynamical  equations,  after  transferring  the  term 
3/3n'2£,  be  replaced  by  the  forms 

i;  -  (1  -  a)  w'l  +    cm'-r)  =  3an'2  F  cos  (pn't  +  q)] 
I  +  (1  +  /3)  n'rj  -  4/3n'2£=  3/3n'2  P  sin  (pn't  +  q)  ]  " 


A  particular  solution  is  £  =  Q  sin  (pn't  +  q),  77  =  Q'  cos  (jm'£  +  q),  provided 
Q'  (~  P'  +  «)    -  Q  (1  -  o)p 


-p--p  =  ..................  (1 

or 

_        Q  Q'  \ 


_ 

a(l+j3)pP'-/3(p*-a)P     ^(1  -  a)  pP  -  a  (p^  +  4/3)> 


In  this  way  any  periodic  terms  on  the  right  of  the  equations  can  be 
represented  by  corresponding  terms  in  |  and  77.  But  the  coefficients  Q,  Q' 
involve  P,  P'  multiplied  by  the  small  quantities  a  or  fi,  and  are  therefore 
extremely  small  unless  A  is  also  very  small.  Now  A=j92(p2  —  1)  when 
a  and  y3  are  ignored  and  therefore,  ceteris  paribus,  sensible  terms  can  be 
obtained  only  when  p  is  very  near  to  0  or  +1. 

Solutions  of  the  same  form  constitute  the  complementary  function  and 
are  determined  by  (17)  when  P  =  P'  =  0.  Then  p  is  given  by 

A  =  p*  -p  (1  -  3y8  -  a/3)  -  4a/3  =  0 
or 

2p*  =  1  -  3/3  -  a/3  +  V{(1  -  3/8  -  a/3)2  +  16a/3} 

It  is  enough  to  retain  in  p  the  terms  of  the  first  order  in  a,  /9,  and  thus 

2p2  =  1  -  3/3-  a/3  ±  (1  -  3/3  -  a/3  +  8a/3) 
so  that  if  ^1,  p2  are  the  two  roots, 


279-281]  Libmtion  of  the  Moon 

Thus  the  periods  of  the  two  possible  terms  are  determined  with  sufficient 
accuracy,  the  former  being  nearly  a  month,  and  if  the  corresponding  co- 
efficients are  Ql}  Q/,  Q2,  Q2',  then  by  (16)  to  the  lowest  order  only 


Hence  a  solution  of  (15)  when  0  is  substituted  on  the  right-hand  side  is 
&  =  Ql  sin  {(1  -  f  /3)  n't  +  q,}  +  Q2  sin  {2  V(-  «£)  t  +  q2} 
%  =  -  Ql  cos  {(1  -  f  £)  n'J  +  ^}  +  2  V(-  £/«)  &  cos  (2  V(- 


and  as  these  expressions  contain  four  arbitrary  constants  Q1}  Q.,,  q1}  q2  they 
represent  the  required  complementary  functions. 

These  arbitrary  terms  again  appear  to  be  insensible.  The  important 
point  is  that  a/8  mast  be  negative,  for  otherwise  the  circular  functions  would 
be  changed  into  hyperbolic  functions  and  the  motion  would  be  unstable. 
This  means  that  (C  —  B)  (A  —  C)  is  negative,  or  again  that  C  is  not  inter- 
mediate in  magnitude  between  A  and  B.  This  is  the  second  condition  of 
stability  which  has  been  found. 

281.     To  terms  of  the  first  order  only, 

L/A=Q,     M/B  =  -3firi2csin(g1  +  '&-n) 
where,  the  secular  inequality  of  the  node  being  taken  into  account, 

g1  +  *r  =  n't  +  6,     ft  =  ft0  -  fin't,     /*  =  +  O004019. 
Thus  in  applying  (17),  P'  =  0,  P  =  —  c,  p  =  1  +  //,,  and  therefore 

(T^»-.  -  (i  -.MI +,.)  °  (i  +  gy^+^+ff^s  -<18> 

If  a,  /3  and  p,  be  regarded  as  small  quantities  of  the  first  order  and  those  of 
the  second  order  be  neglected, 


(19) 
so  that  £  and  77  contain  the  terms 

3/9c  —  38c 


These  terms  contain  the  explanation  of  the  steady  motion  of  the  Moon's  axis, 
which  is  expressed  by  Cassini's  laws. 

For  the  coordinates  of  the  Moon's  pole  of  rotation  relative  to  the  pole  of 
the  ecliptic  may  be  taken  as 

X  =  Q  cos  0  =  |  cos  ((f>  +  ^)  +  V)  sin  (<£  -f  -v|r) 
Y=  6  sin  (f>  —  j;  sin  (0  +  i/r)  —  77  cos  (<f>  +  -\^). 


Libration  of  the  Moon  [CH.  xxm 

Let  the  free  components  fa ,  rj1  be  ignored  and  also  the  forced  oscillations  of 
the  second  order  which  have  still  to  be  found.  Then 

X  =  Q  sin  (gl  +  ts  -  fl  -  <£  -  ^) 

7  =  Q  cos  (gl  +  OT  -  H  -  <f>  -  i/r). 
But  by  (10) 

(f>  +  -*lr  =  yl  +  '&+Tr 
and  therefore 

Z  =  Qsinn,     F=-Qcosfl 

But  the  longitude  of  the  pole  of  the  lunar  orbit  is  H  -  ITT,  so  that  its 
coordinates  are  similarly 

Z'  =  csinfl,      F'  =  -ccosft. 

Hence  these  two  poles  are  always  exactly  on  opposite  sides  of  the  pole  of 
the  ecliptic  provided  Q  is  negative.  This  requires,  since  Q  is  given  by  (19), 
0  >  /3  >  —  I/A.  Hence  0  >  A,  which  is  a  third  condition  to  be  satisfied  by  the 
moments  of  inertia.  The  resultant  of  the  three  places  the  moments  in  the 

order 

C>B>A 

where  C  refers  to  the  axis  of  rotation  and  A  to  that  axis  which  in  the  mean 
is  directed  towards  the  Earth. 

It  is  now  clear  that  the  further  conditions  necessary  in  order  that  the 
second  and  third  laws  of  Cassini  shall  remain  approximately  true  are  one 
and  the  same,  namely  that  those  terms  which  have  been  neglected  in  the 
above  argument  are  really  small  in  comparison  with  Q.  This  quantity  is 
the  mean  value  of  6,  and  its  numerical  value  is  91 ''4  according  to  Franz. 
With  c  =  308'-7  and  /*  =  0'004019  it  follows  that 

-I3  =  (C-A)/B  =  0-000612 

which  should  be  tolerably  well  determined.  It  is  to  be  noticed  that  a,  /3,  7 
are  not  independent,  but  connected  by  the  identity 

a  +  £  +  7  +  a/37  =  0. 

The  product  is  negligible  and  if  7  =  O'OOOS  as  given  above,  then  a  is  of 
exactly  the  same  order  as  7. 

282.  The  terms  of  the  second  order  in  e,  c,  0  can  now  be  found  without 
difficulty,  since  here  it  is  legitimate  to  give  6  and  -v/r  their  values  in  the 
steady  motion.  Thus  0  =  #0,  its  constant  mean  value,  and  since  in  the  steady 
motion  <f>  =  H  4-  ^TT, 

ty  =  g1  +  &  —  O  +  |TT. 

Hence  without  the  terms  of  lower  order  already  treated,  the  expressions  (14) 

become 

L/A=  3<m'2  {e  (00  +  c)  cos  (2#t  +  «r  -  H)  -  e  (00  +  c)  cos  («•  -  fl)} 
M/B  =  3/3-n'2  {-  \e (0'0  +  c)  sin (2^  +  er -  fl)  -  \e  (00  +  c)  sin (tsr -  fl)}. 


281-^83]  Libration  of  the  Moon 

The  corresponding  terms  in  £,  77  can  be  found  in  the  way  explained  in  §  280. 
But  since  OT  and  H  change  slowly  p  is  nearly  2  in  the  case  of  the  terms  which 
contain  2g1  in  the  argument.  Their  counterpart  in  £,  77  is  therefore  negligible. 
With  the  other  pair  p  is  very  small.  The  secular  changes  in  the  node  and 
perigee  may  be  expressed  by 


so  that  p  =  IJL  +  v,  and  2P  =  P'  =  —  e  (00  +  c).     Hence  (17)  give 

Q  __  q 

2cT(l  ~ 


(p*  -  a)  (p2  +  4/3)  -  (1  -  a)  (1 

which,  when  simplified  by  the  removal  of  all  but  the  most  significant  quantities 
in  the  denominators,  become 


The  terms  of  the  second  order  are  therefore  simply 


sin(ts-  -  H),     rj3  =  \$e  cos(^  -  O)  ......  (21) 

fj,  -}-  f 

Now  i;  =  0-008455,  /i  +  v  =  1/80  nearly,  and  00  +  c  =  400'.  Also  e  =  0'0549 
and  with  the  above  values  of  a  and  /3,  3«<?  =  -  \$e  =  0*00005.  Hence  both 
coefficients  are  numerically  l/-6,  and 

&  -  r-6  sin  (tsr  -  fl),     %  =  -  l'-6  cos  («•  -  H) 
the  period  being  80  lunar  months  or  6  years. 

283.     When  the  several  'terms  found  are  combined, 

£  =  £  +  &  +  &,     ^  =  % 
and  by  (9) 

Wj  =  T)  -  ft)3£,     <y2  =  I 
Now  with  the  approximate  forms  (20) 

%2=-rir).2,     r)^  = 
and  from  (21) 

^3  =  n(/j,  +  v)r}3,     7}3=- 

Hence,  putting  &>3  =  n    here  and  neglecting  the  arbitrary  terms  £1}  %,  the 
existence  of  which  has  not  been  established  by  observation, 


and  (yu,  +  v)  is  relatively  unimportant  here. 

One  remark  is  necessary  however.     For  the  sake  of  simplicity  and  in  order 

to  concentrate  attention  on  the  main  feature  of  the  motion,  the  coefficients 

of  £2  and  ?;2  in  (20)  were  made  numerically  equal  by  the  simple  expedient 

of  neglecting  /u.2(=  0'000016)  in  comparison  with  fi.     Consistently  with  this 

p.  p.  A.  21 


322  Libration  of  the  Moon  [OH.  xxni 

the  factor  (1  +  /&)  has  been  omitted  in  finding  |2>  fy,  and  the  result  is  that 
£>>  %  do  not  appear  in  a>1;  <o.2.  This  factor  can  only  be  reinstated  correctly 
after  /*2  has  been  restored  in  £2,  i)2.  Now  by  (18)  £,,  rj..  are  of  the  form 

£,  =  {(1  -f  /tt)2  —  «}  G  sin  g.     ?72  =i  —  (1  —  a)  (1  +  //,)  (?  cosg 
where  g  =  gl  -f  w  —  O.     Hence 

|2/7i'  =  (l  -f  /A)  {(l+yu,)2-aj  Gcosg  » 

%/w'  =  (1  4-  jtt)2  (1  —  a)  (r  sin  # 
and  the  contributions  to  wl,  co2  are  given  by 

Awj/w'  =  —  a  (2/z  +  /ti2)  (r  sin  g 

Aft>2/w'  =  (!  +  /*)  (2/i  4-  /i2)  6s  cos^r. 

The  factor  a  shows  that  Ae^  is  very  small  and  if  yf  as  well  as  a  be  now 
rejected, 


Hence  in  a  numerical  form  the  forced  rotations  are  finally  given  by 

mjn'  =  -  £,  =  -  r-6  sin  (w  -  II) 

.    a>2fn'  =  i)3  —  2/j,r)2  =  —  1''6  cos  (-57  —  O)  —  0'*7  cos  (gl  +  CT  —  O) 
since  G  =  -  91H  and  /*  =  0*004. 

With  the  more  exact  expressions  the  coefficient  in  £,  is  numerically 
greater  than  that  in  i]2,  the  difference  being  —  //,(!+//,  + a)  (7  or  —  pG.  This 
amount,  22",  may  be  divided  equally  between  the  two  coefficients  without 
disturbing  the  observed  mean  inclination  of  the  lunar  equator  to  the  lunar 
orbit,  and  thus 

£•2=-  91''6  sin  (g1  +  •&  -  n),     772  =  91''2  cos  (gl  +  o  -  H). 
Lastly,  by  (7),  if  ^2  the  free  libration  in  longitude  be  ignored, 

/  ,  Oil  »  0-000242 

«>=      -W-      -  073-3— >  •  7  COB  ^  +  Q.OQ1865  _  y/ .  7  cos  0 

where  the  coefficients  are  expressed  in  circular  measure.  Thus  the  position 
of  the  instantaneous  axis,  relative  to  the  principal  axes  of  the  Moon, 

x\  wl  =  yj  6>2  =  zj  o)3 

is  determined.  It  has  therefore  been  seen  under  what  conditions  Cassini's 
laws  are  approximately  true,  and  how  far  they  must  necessarily  be  modified 
by  disturbing  actions. 

• 

The  latest  results  from  observation,  by  M.  Puiseux  of  Paris,  seem  to  be 
at  variance  with  the  foregoing  theory.  It  is  probable  that  it  will  be  necessary 
to  treat  the  Moon  as  a  deformable  body,  as  the  observed  variations  of  latitude 
have  shown  to  be  requisite  in  the  case  of  the  Earth.  The  above  theory  is 
very  largely  due  to  Poisson. 


CHAPTER   XXIV 

FORMULAE    OF   NUMERICAL    CALCULATION 

284.  If  we  consider  a  function  of  one  variable  or  argument  only,  for  the 
sake  of  definiteness,  it  can  be  represented  in  three  distinct  ways,  namely : 

(1)  By  an  analytical  form,  e.g.  sin  a;  or  a  hypergeometric  series  F  (a,  /3,  7,  as). 
The  effectiveness  of  such  a  form  depends  on  the  knowledge  of  its  properties 
and  the  facility  with  which  it  submits  to  the  ordinary  operations  of  mathe- 
matics. 

(2)  Graphically,  by  a  curve.     This  gives  a  continuous  representation. 
Values  of  the  function  corresponding  to  particular  values  of  the  argument 
can  be  obtained  and  the  processes  of  differentiation  and  integration  can  be 
performed  mechanically.     But   the   accuracy  of  the   results   is   limited   in 
practice. 

(3)  Numerically,  by  a  series  of  isolated  values.     This  gives  a  discon- 
tinuous representation,  but  one  capable  of  very  great  accuracy.     In  theory 
this  does  not  serve  to  define  the  function,  for  it  may  vary  in  any  manner 
between  the  given  values.     Even  in  practice  the  representation  does  not 
cover  terms  in  the  function  with  a  period  of  the  same  order  as  the  intervals 
between   the  values.     But  with  due  care  this  limitation  causes  little  in- 
convenience. 

Each  mode  of  representation  has  distinct  advantages  of  its  own  and  to 
pass  from  one  to  another  is  a  problem  frequently  arising  and  often  attended 
by  great  difficulty.  The  form  (1)  may  be  considered  the  ultimate  expression 
of  natural  truth,  but  it  has  no  absolute  superiority.  Thus  integration  may 
be  practically  impossible  in  this  form  and  must  be  replaced  by  a  mechanical 
quadrature. 

A  function  determined  by  a  series  of  observations  or  experiments  falls 
generally  under  the  form  (3).  Now  the  variable  quantities  which  occur  in 
Astronomy,  e.g.  the  coordinates  of  the  Moon,  are  in  general  so  complicated, 
even  when  an  expression  in  analytical  form  is  available,  that  for  practical 
purposes  it  is  necessary  to  use  an  ephemeris,  or  a  table  of  values  calculated 
for  equal  intervals  of  time  (not  necessarily  one  day,  as  the  name  would 
imply).  It  is  therefore  necessary  to  consider  how  functions  represented  in 

21—2 


324  Formulae  of  Numerical  Calculation       [OH.  xxiv 

this  way  may  be  manipulated  so  as  to  give  intermediate  values  by  inter- 
polation for  comparison  with  the  results  of  observation,  and  also  to  render 
numerical  differentiation  and  integration  possible. 

285.  Let  w  be  the  constant  interval  of  the  argument  and  2/n==/(«  +  nw) 
be  the  function  to  be  considered,  the  values  of  yn  being  given  for  consecutive 
integral  values  of  n.  A  simple  difference  table  can  be  formed  thus : 

a  +  (n  —  l)w 

a  +  nw 

• 

a  +  (n  +  \)w 
Now  let  two  operators  A,  8  be  introduced  such  that 

Then  it  follows  that 

ASyn  =  A  (yn  -  yn^)  =  yn+l  -  2yn  +  yn-i  =  S  (yn+i  -  y»)  = 
Hence  the  operators  A,  8  are  commutative,  and  similarly  it  is  easily  seen 
that  they  obey  all  the  laws  of  ordinary  algebra.  The  inverse  operators 
A-1,  8~l  may  be  defined  so  that  A  A"1  =  1,  8B~l  =  1.  Then  the  table  of 
differences  may  be  replaced  by  a  table  of  operations  which,  acting  on  yn,  will 
reproduce  the  difference  table,  thus  : 

L^        G  O 

8 

1  AS 

A 

AS-1  A2 

The  two  operators  are  not  independent,  for  the  position  of  AS  in  this  table 
shows  that  they  are  connected  by  the  homographic  relation 

Let  x  be  the  variable,  so  that  y  =/(#),  and  let  D  =  djdx.     Then 


or  1  +  A  =  ewD.     Hence 


Thus 


=  (1  +  wD  +  i2wn-D2+  . 

=  elvD  ,f(x}  ..........................................  (2) 


=f(x)  +  qwf  (x}  +  Iftff"  (x)  +  .  .  . 
=  f(x  +  qw). 


284-286]  Formulae  of  Numerical  Calculation  325 

which  is  Newton's  original  formula  of  interpolation  and  can  be  written  in 

the  form : 

(  /~\  \ 

» (3) 


where  j  q    by  a  proper  choice  of  n  may  always  be  taken  <  ^,  and  in  any.  case 
should  not  exceed  1.     The  coefficients  are  simple  binomial  coefficients. 

286.  The  differences  A,  A2,...  are  diagonal  differences  in  the  table. 
But  the  most  useful  formulae  involve  central  differences,  lying  on  or  adjacent 
to  a  horizontal  line  in  the  table.  If  the  blank  spaces  in  the  odd  columns  are 
filled  by  the  arithmetic  means  of  the  entries  immediately  above  and  below, 
the  operators  in  the  complete  central  line  are 

1         KA  +  S)         AS         £(A  +  S)AS         (AS)2      ... 
which  can  also  be  written,  by  introducing  two  new  operators  K,  k, 

I  k  K  kK  K* 

where 


.(4) 


Thus  k  cannot  be  expressed  rationally  in  terms  of  K,  and  in  order  to  find  a 
formula  in  terms  of  central  differences  it  is  necessary  to  expand  in  terms 
of  K,  keeping  only  the  first  power  of  k.  Thus 

kuq  +  vq  ................  -.....(5) 


where 

uq  = 

v  = 


It  is  easily  verified  that 

uq  (1  +  PO  +  VQ  =  uq+1,    uq  (K  +  ±K*)  -f  vq  (1  +  %K)  =  vq+l 


since 

*"  vr-1 
Also 


-  2   i  <«  -  2r)      .   +  (r  +  l)  (1  +  i^)'—'  (K  + 

=  2  J, !(«-/)  +  (|.;\)}  (1  + 

»-  (K 


326  Formulae  of  Numerical  Calculation       [OH.  xxiv 

It  is  therefore  possible  to  write 


vq  =  1  +  q2brKr,     uq  =  q±22(r+l)  br+1  Kr. 

Let  br  become  brf  mvq+1,  uq+1,  and  equate  the  coefficients  of  Kr~l  in  the  first, 
and  of  Kr  in  the  second,  recurrence  formula.     Thus 

2rbr'  =  2rbr  +  (r  -  1)  &.,_>  +  qb^ 
(q  +  l)  br'  =  2rbr  +  |  (r  -  1)  b^  +  qbr  +  $qbr^ 

and,  on  eliminating  &/, 

2r  (2r  -  1)  br  =  (q  +  r  -  1)  (q  -  r  +  1)  &,_,. 

This  shows  that 


where  JL  is  a  constant,  and  since  b^  —  \q,  A  =  1.     Hence 

-)^  .........  (6> 

and  the  first  terms  of  the  complete  formula  are  therefore 


...y.  ......  (7) 


This  series  was  found  by  Newton,  but  is  generally  known  as  Stirling's  formula. 
It  is  here  taken  as  fundamental,  and  other  results  are  deduced  from  it. 

287.     The  formula  of  Gauss  depends  on  the  even  central  differences  and 
the  odd  differences  of  the  line  below,  the  operators  being  therefore 

1  K  K2 

A 

These  are,  in  terms  of  k,  K, 


But  (5)  may  be  written  in  the  form 
(l  +  ^  =  (k  +  ^)uq 
where  by  (6) 

r,  =  ,,  -  $KU,  -  1  +  s 


_ 


280-288]  Formulae  of  Numerical  Calculation  3*27 

This  gives  the  coefficients  of  the  even  central  differences,  the  coefficients  of 
the  odd  differences  of  the  adjacent  line  being  still  given  by  uq.  The  first 
terms  of  the  complete  formula  are  therefore 


O  ! 


If  the  order  of  the  difference  table  were  reversed,  —  8  would  take  the  place  of 
A  and  the  sign  of  w  would  be  changed.     Hence  similarly 

i,  ......  (10) 


By  choosing  either  (9)  or  (10)  q  can  always  be  taken  between  0  and  +  £. 

288.  The  formula  of  Bessel  contains  the  odd  differences  in  the  line 
immediately  below  the  central  function,  with  the  mean  even  differences 
of  the  same  line,  so  that  the  operators  are 

1+£A,     A,     (1+iA)^,     &K,     (1  +  £A)#2,  .... 

The  odd  differences  are  thus  the  same   as   in  the  formula  of  Gauss,  and 
therefore 

=  Aw  +  F  =  l 


where,  by  (6)  and  (8), 


-  l 

(ll) 


This  gives  the  coefficients  of  the  odd  differences,  and  the  coefficients  of 
the  even  (mean)  differences  are  given  by  F9.  Hence  the  first  terms  of  the 
complete  formula  are 

^  *       — 

yn+q  — 


Bessel's  own  form  differs  from  this  in  the  first  two  terms,  being  written 


328  Formulae  of  Numerical  Calculation        [CH.  xxiv 

which  is  of  course  equivalent,  but  is  not  symmetrical  with  respect  to  the 
middle  of  the  tabular  interval.  To  make  this  symmetry  clearer,  let  p  +  ^  be 
substituted  for  q  in  (12),  which  then  becomes 

+  |A)  +p  .  A  +  tp*  .  (1  +  JA)  K  +p  .  P' 


_. 

When  the  sign  of  p  is  reversed,  the  terms  of  even  order  are  unchanged  and 
the  terms  of  odd  order  are  simply  reversed  in  sign.  If  terms  of  the  two 
orders  are  computed  separately,  two  interpolations  —  corresponding  to  I  p  — 
are  obtained  at  the  same  time.  This  is  of  great  advantage  in  systematic 
interpolation  to  regular  fractions  of  the  tabular  interval,  e.g.  in  reducing  the 
12-hourly  places  of  the  Moon  to  an  hourly  ephemeris.  Stirling's  formula 
presents  a  similar  advantage.  But  (13)  becomes  particularly  simple  at  the 
middle  of  an  interval,  for  then  q  =  %  or  p  =  0,  and  the  odd  differences  dis- 
appear. Thus 


-•n  '+...}?»  .........  (14) 

and.  this  gives  intermediate  values  with  great  ease  and  accuracy. 

289.  When  the  values  of  a  function  y  are  known  only  at  irregular 
intervals  of  the  argument  x,  as  in  an  ordinary  series  of  observations,  the 
function  is  strictly  indeterminate  in  the  absence  of  other  information  as  to 
its  form.  Nevertheless,  when  n  values  ylt  ...,yn  are  known,  corresponding 
to  aslt  ...,  xn,  a  formula 

y  =  a0+a1a;+  ...  +  a«_]  xn~l 

can  be  found  which  is  satisfied  by  the  n  values  and  within  the  interval 
a?i  to  xn  will  generally  resemble  the  true  function  closely.  The  n  coefficients 
can  be  determined  by  the  linear  equations 


(r  =  1,  .  .  .  ,  n).    These  can  be  solved  in  the  ordinary  way,  but  it  is  immediately 
obvious  that  the  result  can  be  written 


•  =\  .  .  .  n 

(Xr       Xi)  .  .  .  (Xr  —  Xn) 

where  the  numerator  of  the  fraction  written-  does  not  contain  (x  -  #,.).  For 
this  equation  becomes  an  identity  when  xr>  yr  are  substituted  for  x,  y.  The 
expression  on  the  right  is  a  polynomial  of  degree  n  —  1  in  x  and  the  equation, 
since  it  is  satisfied  by  every  pair  (xr,  yr),  must  be  identical  with  the  previous 
equation,  the  coefficients  in  which  can  be  written  down  by  comparison.  The 
formula  (15)  is  due  to  Lagrange  and  is  directly  suitable  for  interpolation, 


288-290]  Formulae  of  Numerical  Calculation  329 

differentiation  and  integration.  An  illustration  of  its  use  in  a  case  where 
n  =  3  has  been  given  in  §  71.  When  n  is  large  the  formula  naturally  be- 
comes inconvenient  for  practical  purposes. 

290.  Returning  to  the  function  with  known  values  at  regular  intervals 
of  the  argument,  let  us  consider  the  process  of  mechanical  differentiation. 
By  (2) 

wD    =  log(l  +  A)      =  A-iA2  +  iA3-...  I 

w2Z>2  =  {log(l  +  A)}2  =  A2  -  A8  +  1JA4  -.../' 

These  formulae  are  suitable  only  in  simple  cases  where  great  accuracy  is  not 
required.  The  loss  of  accuracy  is  a  natural  tendency  when  differentiation  is 
concerned.  The  forms  (16)  also  apply  only  to  the  tabulated  value  of  the 
argument.  But  since 

x  =  a  +  (n  4-  q)  w,     wD  =  wdfdx  =  d/dq 

a  formula  of  differentiation  can  be  derived  from  every  formula  of  interpolation. 
Thus  Bessel's  formula  (12)  gives 

'- 

and  analogous  forms  may  be  derived  similarly  by  differentiating  (7)  and  (9) 
with  respect  to  q. 

But  there  are  some  particular  cases  of  special  simplicity  and  importance 
in  the  formulae  of  central  differences.  According  to  (6)  uq  is  an  odd  function 
and  vq  an  even  function  of  q.  Now  when  9  =  0,  d/dq  is  the  coefficient  of  q 
and  dz/dq2  is  twice  the  coefficient  of  <£  in  kuq  +  vq.  These  coefficients  can 
easily  be  taken  from  kuq  and  vq  respectively,  and  give,  by  (6)  or  (7), 


7! 


......  (18) 


and 


21  (19) 

Both  (18)  and  (19)  involve  the  alternate  differences  in  the  central  tabular 
line. 

Similarly  when  Vq,  Uq  are  expressed  in  terms  of  p  =  q  +  |  instead  of  q  as 
in  (8)  and  (11),  Vq  is  an  even  function  and  Uq  is  an  odd  function  of  p. 
WThen  q  =  i,p  =  0  and  d/dq  is  the  coefficient  of  p  and  d*/dq*  is  twice  the 


330  Formulae  of  Numerical  Calculation        [on.  xxiv 

coefficient  of  p-  in  (1  +  -1A)  Vq  +  A  Uq.     These  coefficients  can  readily  be  taken 
from  (13),  which  sufficiently  indicates  the  law  of  formation,  and  thus 


and 


-  (32 .  52 .  72  +  I2 .  52 .  72  +  I2 .  32 .  72  + 12 .  32 . 52) 


The  distinction  between  the  operators  (1  +  A)4  and  (1  +  £A)  must  be 
carefully  noted.  That  on  the  left,  (1  +  A)",  indicates  an  addition  of  half  the 
tabular  interval  to  the  argument,  so  as  to  apply  the  differentiation  at  the 
right  point,  which  is  the  middle  of  the  interval.  That  on  the  right,  (1  +  ^A), 
merely  denotes  the  mean  of  adjacent  differences  in  a  vertical  column  of  the 
difference  table. 

291.  Convenient  methods  for  mechanical  integration  or  quadrature  can 
now  be  deduced.  The  formulae  for  differentiation  just  found,  (18),  (19),  (20), 
(21),  are  of  the  form 

wD  =  kSl  (K),  w*D*  =  S2  (K) 

wD  (1  +  A)4  =  AS3  (K),    w*D2  (1  +  A)4  =  (1  +  \  A)  S,  (K) 
S  (K)  denoting  a  power  series  in  K.     Hence 

w-1  D-1  =  k-^/S,  (K),  w-*  D-2  =  1  fSa  (K) 


The  coefficients  of  the  reciprocals  of  the  K  series  must  be  expressed  more 
appropriately,  thus  : 


=  kite  =  k(K+ 


+ 


(1  +  A)  (1  +  ^A)-1  =  (1  +  £A)  {1  +  |A2  (1  +  A)-1}-1  =  (1  +  £A)  (1  +  J 


It  is  therefore  necessary  to  multiply  -8-^  and  $4  by  (1  +  \K)  before  finding  the 
reciprocals  of  the  series  by  division  in  order  to  have  results  for  D~l,  Z)~2  of 


290, 291  ]  Formulae-  of  Numerical  Calculation  331 

exactly  the  same  form  as  those  already  found  for  D,  D-.     These  results  are 
easily  found  to  be 


(22) 

(23) 

)    (24) 

:•  +...)..  .(25) 

The  development  is  here  carried  as  far  as  differences  of  the  fifth  order. 
This  is  generally  sufficient. 

It  is  now  necessary  to  examine  the  meaning  of  these  purely  formal 
results.  The  operator  K,  like  its  components  A,  8,  is  such  that  KK~l  =  1, 
and  therefore,  as  K  represents  a  move  two  places  to  the  right  in  the  table, 
K~l  represents  a  move  two  places  to  the  left.  The  difference  table  now 
requires  an  extension  not  hitherto  contemplated,  and  the  central  line  of  the 
table  of  operators,  with  the  adjacent  lines  above  and  below,  now  becomes : 

S  SK  SK*   ... 

[k]        K  [kK]        K*          [kK*}  ... 

A  ra 


Here  1  corresponds  to  the  original  entry  yn  in  the  table.  The  natural 
differences  as  directly  formed  are  expressed  simply,  while  those  which  are 
means  of  the  entries  immediately  above  and  below  are  enclosed  by  [  ]. 
But  while  the  symbols  occurring  in  the  columns  to  the  right  of  the  central 
column  (representing  the  function  itself)  will  be  readily  understood,  the 
construction  of  the  columns  to  the  left  must  now  be  explained.  The  numbers 
in  the  first  column  to  the  left  are  such  that  their  differences  appear  in  the 
central  column.  Thus 


-  Mr-')  yn  =  yn>     A*"-'  yn  =  yn  +  SK~>  yn 

and  when  one  number  in  this  column  is  fixed,  the  rest  are  formed  by 
adding  successively  (when  proceeding  downwards)  the  tabulated  values  of 
the  function.  The  entries  in  this  column  therefore  contain  an  additive 
arbitrary  constant.  The  second  column  to  the  left  is  related  to  this  first 
column  in  exactly  the  same  way  as  the  first  column  to  the  central  column, 
and  therefore  contains  another  arbitrary  constant,  but  is  otherwise  definite. 

The  use  of  four  different  operators  in  the  table  may  seem  excessive,  since 
they  are  all  expressible  in  terms  of  one.     In  fact 


and  this  suggests  another  mode  of  development  which  has  here  been  de- 
liberately avoided.     But   all  these  operators  have  simple  special  meanings 


332  Formulae  of  Numerical  Calculation        [CH.  xxiv 


and  it  is  important  to  notice  that  k8~*  and  (1  +  |A)  are  equivalent,  but  quite 
distinct  from  AAr1,  though  in  the  complete  table,  in  which  the  mean  differ- 
ences are  filled  in,  they  all  three  denote  one  vertical  step  downwards. 

292.  As  with  A"1  and  the  other  operators,  D"1  is  such  that  DD~l  =  1,  or 
D,  D~l  represent  inverse  operations.  And  since  D  represents  differentiation, 
Dr*  represents  integration.  Thus  take  the  formula  (24).  The  column  AAr1 
being  formed  with  an  arbitrary  constant,  the  right-hand  side  of  the  equation, 
operating  on  yn,  will  produce  a  function  (represented  in  tabular  form)  which 

is  w~l  D~*  (1  +  A)*  yn  =  w~l  D~l  yn+<-  On  the  application  of  D  or  differentia- 
tion, this  becomes  w~1yn+,.  Hence  the  meaning  of  the  formula  is 


where  m  is  written  for  n  +  £.  The  lower  limit  is  arbitrary.  But  the  right- 
hand  side  also  contains  an  arbitrary  constant,  and  this  constant  can  now  be 
chosen  so  as  to  fix  the  lower  limit  of  integration.  For  let  this  limit  be 
a  +  ^w.  If  then  ra  =  £,  w  =  0  in  (26) 

O^Atf-'  +  ^A-^Atf  +  Tn/^Aff--...)^    ......  (27) 

and  the  value  of  A/iT"1  .  y0  is  now  determined.  With  it  the  whole  of  the 
corresponding  column  can  be  definitely  calculated  by  successive  additions  of 
the  values  of  the  function.  When  this  is  done,  (26)  represents  the  definite 
integral  of  y  between  the  limits  a  +  \w  and  a  +  (n  +  ^)  w. 

Quite  similarly  the  meaning  of  (22)  is  seen  to  be 

ra+nw 


ydx  =  (kK-*  -&k  +  7-VV  kK  -  itffa  kK'  +  ...)yn..  .(28) 
where  the  lower  limit  is  a  when 


But  the  latter  form  is  not  convenient,  because  kK~l  y0,  which  is  hereby  deter- 
mined, is  the  mean  of  two  numbers  not  yet  known.     Now 


2/0  =  Atf-1  y,  +  8K->  y0,     y,  -  AIT"  y,  -  SK~*  y0 

and  therefore 

dfcskK*-...)yt  .........  (29) 


Thus  A-K""1  .  y0  is  determined,  and  the  calculation  proceeds  as  in  the  previous 
case.  It  is  to  be  noticed  that,  though  (27)  has  been  derived  from  (26)  and 
(29)  from  (28),  (26)  can  be  used  in  conjunction  with  (29),  giving  a  and 
a  +  (n  +  £)  w  as  the  limits  of  integration,  or  (28)  with  (27),  giving  a  +  nw  as 
the  upper  limit  and  a  +  ^w  as  the  lower  limit. 


291-294]  Formulae  of  Numerical  Calculation  333 

293.     In  a  similar  way  (23)  and  (25)  give  the  second  integrals,  thus 

ra+nw  r  rx 

K*-...}yn  ......  (30) 


r 
J  b 


F    r 
[J 


............  (31) 

where  m  =  n+  \  as  before.  The  lower  limit  c  of  the  subject  of  the  second 
integration  is  arbitrary.  But  if  the  first  summation  column,  on  the  left  of 
the  function  y,  has  been  based  on  (29),  c  =  a  ;  if  it  has  been  based  on  (27), 
c  =  a  +  ^w.  The  lower  limit  b  of  the  second  integration  is  also  arbitrary  and 
corresponds  with  the  additional  arbitrary  constant  in  the  second  summation 
column  K~\  The  latter  is  easily  determined  by  taking  the  case  b  =  a,  n  =  0 
of  (30).  Thus 

0  =  (^  +  TV-^o^  +  ^V8o^2--.-)2/o  ...............  (32) 

This  gives  K~ly0,  and  the  whole  of  the  second  summation  column  becomes 
determinate  when  the  first  column  has  been  fixed.  Or  again,  if  the  lower 
limit  b  is  to  be  a  +  \w,  (31)  gives  when  b  =  a  +  ^w,  m  =  %,  n  =  0, 


or 


This  is  quite  general  whatever  the  value  of  c,  or  of  A.fif"1  y0,  may  be.     But  as 
c  —  b  usually,  (27)  can  be  used  in  this  case,  and  then 

K-1  2/o  =  (A  (1+  A)  -  jti»  (3  +  2A)  K  +  jrfMn  (5  +  3  A)  K*  -  .  .  .}  y0  .  .  .(34) 
When  the  second  summation  column  is  based  on  (34)  and  the  first  on  (27) 
x  =  a  +  \w  is  the  common  lower  limit  for  the  double  integration.  When 
(29)  and  (32)  are  used  in  forming  these  columns,  x  =  a  is  the  common  lower 
limit.  In  either  case  (30)  and  (31)  give  the  values  of  the  double  integrals 
to  the  upper  limits  x  =  a  +  nw  and  x  =  a  +  (n  +  -|)  w  respectively. 

No  attention  has  been  given  here  to  the  limitations  of  the  method  which 
are  imposed  by  the  conditions  of  convergence  of  the  expansions  employed. 
In  general  the  question  is  settled  in  practice  by  obvious  considerations.  But 
for  a  critical  estimate  of  the  accuracy  attainable  it  is  clearly  important. 

294.  There  is  also  a  trigonometrical  form  of  interpolation,  otherwise 
known  as  harmonic  analysis,  which  is  of  great  importance.  This  is  intimately 
related  to  Fourier's  series,  and  indeed  amounts  to  the  calculation  of  the 
coefficients  of  this  expansion.  It  will  be  well  to  recall  the  principal  pro- 
perties of  the  series,  which  may  be  stated  thus  : 

The  sum  of  the  infinite  series 

cr.0  +  2  (an  cos  nx  +  bn  sin  nx) 


334  Formulae  of  Xnuierical  Calculation         [CH.  xxiy 

(n  a  positive  integer),  where 

1    /"2ir  1   i"2ir  1    f2jr 

a0  =  o~  I    y  (^)  ^j     an  =  —       f(x)  cos  w#  ^>     bn  =  —  I    f(x)  sin  ?w;  e&c 
An  j  Q  TJO  7!"^o 

\&f(x)  throughout  the  interval  0  <x<  2?r,  provided  f(x)  is  continuous. 

At  any  point  so  in  the  interval  where  f(x)  is  discontinuous,  the  sura  of 
the  series  is  |  [f(x  -  0)  +f(x  +  0)}. 

It  is  assumed  that  the  number  of  finite  discontinuities  and  the  number  of 
maxima  and  minima  of  f(x)  are  finite.  These  conditions  are  more-  than 
sufficient  and  are  always  satisfied  by  the  empirical  functions  of  practical 
computation. 

The  expansion  is  unique  in  the  sense  that  no  other  coefficients  can  make 
the  given  series  represent  the  same  function  over  the  stated  interval  so  long 
as  n  remains  integral. 

If  the  series  is  absolutely  convergent  for  all  real  values  of  x  it  is  also 
uniformly  convergent.  Its  sum  has  then  no  discontinuities  and  has  the 
same  value  at  x  =  0  and  x  =  2?r. 

The  sum  of  the  series  is  a  periodic  function,  with  the  period  2?r.  If  f(x) 
is  also  periodic  with  the  same  period,  it  coincides  with  the  sum  of  the  series 
for  all  values  of  x,  but  otherwise  the  functions  coincide  only  in  the  interval 
0  <  x  <  27T.  If  f(x)  =/(—  x)  =f(x  +  2?r),  f(x)  is  represented  by  a  Fourier 
series  containing  cosine  terms  only  (bn  =  0).  If  f(x}  =  — /(—  x)  =f(x  +  2-n-), 
f(x)  is  represented  completely  by  a  series  containing  sine  terms  only 
(a0  =  an  =  0).  Similarly  an  arbitrary  function  can  be  represented  within 
the  interval  0  to  TT  either  by  a  sine  series  or  by  a  cosine  series  when  one  of 
the  functions  +/(27r  —  x)  is  assigned  to  the  interval  TT  to  2?r. 

295.  When  the  function  is  given — and  the  term  function  has  here  an 
exceptionally  wide  meaning — the  coefficients  in  its  expression  as  a  Fourier's 
series  can  be  calculated  by  a  special  kind  of  integrator,  known  as  an  Harmonic 
Analyser,  of  which  several  forms  have  been  invented.  But  here  the  equivalent 
arithmetical  processes  will  be  considered. 

When  the  function  is  represented  by  a  definite  number  of  distinct  values 
it  is  obvious  that  only  a  finite  number  of  terms  in  the  series  can  be  deter- 
mined, and  it  is  necessary  to  assume  that  the  practical  convergency  of  the 
series  is  such  that  the  remainder  after  a  certain  point  is  negligible.  Let  the 
finite  series  be 

n 

u  =  a0+  2  (cti  cos  id  +  bf  sin  id} 

i  =  l 

with  2w  +  1  corresponding  pairs  of  values,  u  =  u,.,  6  =  6r.     From  the  linear 
equations 

ur  =  a0  +  2  (a>i  cos  idr  +  bi  sin  i0r) 


294-296]  Formulae  of  Numerical  Calculation  335 

the  coefficients  a0,  a;,  6;  can  be  found  in  the  ordinary  way.  It  is  also  easy  to 
represent  the  result  .by  a  formula  analogous  to  Lagrange's  formula  of  inter- 
polation (15).  But  when  6r  =  2r?r/(2n  +  1)  the  solution  can  be  effected  in  a 
very  simple  way. 

It  is  necessary  to  consider  the  sums'  of  two  very  simple  series.  In  the 
first  place 

s-l  s-l 

2  sin  ra  =  2  {cos  (r  —  |)  a  —  cos  (r  +  £)  a} /2  sin  ^a 

r=Q  0 

=  {cos^a  —  cos(s  —  i)a}/2  sin^a 
=  sin  ^sa sin  |(s  —  1)  a/sin ^a 

and  this  is  0  if  a  =  2pfr/s.  Even  when  p  =p's,  p  and  p'  being  both  integers, 
and  therefore  sin  |  a  =  0,  this  remains  true,  for  every  term  of  the  series  is  then 
zero.  Similarly 

s-l  *-l 

2  cos  ra  =  2  {sin  (r  +  ^)a  —  sin  (r  —  -£)  «}  /  2  sin  \  a 

r=0  0 

=  {sin(s—  ^)a  +  sin  £a}/2  sin^a 

=  sin^sacos£(s  —  1)  a/sin  £a 

and  this  is  0  also  if  a  =  2pTr/s,  unless  p  =p's.  In  the  latter  case  each  term  of 
the  series  is  1  and  the  sum  is  s.  Thus  both  the  series  vanish  for  a  =  2pTr/s, 
except  the  cosine  series  when  a  =  Zp'jr. 

296.  Let  u  =  ur  be  the  value  of  the  function  corresponding  to  the  value 
of  the  argument  6  =  ra..  The  series  will  not  now  be  limited  to  a  finite  number 
of  terms.  Then 

s-l 

S  ur  cosjra  =  «02  cosjra  +  S  2  (a,;  cos  jra.  cos  ira.  +  hi  cosjra  sin  ira) 

r=0  r  i  r 

=  a02  cosjra  +  %  2  2  «;  {cos  (i  +j)  ra  +  cos  (i  —j)  ra} 


r=0 


ur  sin  jra  =  «02  sin  jra  +  2  2  (at  sin  jra  cos  ira  +  6t-  sin  jra.  sin  ira) 

i   r 

{cos  (i  —j)  ra  —  cos  (i  +j)  ra} 


when  a  =  ZTT/S,  for  all  the  sine  terms  vanish  immediately  in  the  sum  with 
respect  to  r.  The  cosine  terms  also  vanish  in  the  sum  unless  j,  i+j  or  i  —j  is 
a  multiple  of  s  (including  zero).  Thus,  j  having  in  succession  all  values  from 
1  to  £  (s—  1),  or 


.(35) 


336  Formulae  of  Numerical  Calculation        [CH.  XXIY 

When  s  equidistant  values,  u0,  ...,  iig^,  (u8  =  'uo)>  ai*e  known  the  operations 
indicated  on  the  left  are  easily  performed.  Then,  if  the  series  converges  so 
rapidly  that  the  higher  coefficients  can  be  neglected,  a,n ,  alt  bl,  ...  are  deter- 
mined, as  far  as  aj  (,_!>,  6j(g_D  if  s  is  odd,  and  as  far  as  a^s,  6ig_x  if  s  is  even. 
The  lower  coefficients  will  naturally  be  calculated  much  more  accurately  than 
the  higher,  for  there  is  little  reason  to  suppose  a^s+l  small  in  comparison  with 
a^g-i.  But  it  is  well  to  compute  the  higher  coefficients  as  a  practical  test  of 
convergence. 

297.  It  is  usually  convenient  to  make  s  an  even  number,  and  indeed  a 
multiple  of  4,  so  as  to  divide  the  quadrants  symmetrically.  Let  s  =  2n  and 
let  the  terms  of  higher  order  than  an,  6M_,  be  neglected.  Then  (35)  become 

1  2M~1  1  „  jr-rr       7       1  v       .   jr-jr 

«„  =  —    z,  ur.     a-,  =  -  2,ur  cos^ — ,     ft,- =  -  zwr  sm- —  (do) 


2w  r=0    '  n "  n   '  n  n 

(j  =  l,  2,  ...,n-l).     When  j  =  n, 

so  that  an  is  determined,  but  not  bn\  and  this  is  natural,  for  2n  coefficients  i 
addfbion  to  a0  cannot  be  derived  from  2n  values  ur. 

Let  n—j  be  written  for  j  in  (36).     Then 

1  n^,1  (  ?VTT\      1  ^,  ,  ir?r 

an_;  =  -    2,  wr  cos  I  rtr  —  —     }  =  -  2,  (—  LY  ur  cos- 
n  ,.=0  \  w  /     |»  w 

7           1  <        •    /         Jr7r\         !  v  /  •   Jr7r 

&„_,•  =  -  S  wr  sm    r?r  — *-  -     = z  (—  1 )'  ur  sm  - — . 

n  \  n  )         n  n 

Hence 


ence 

1  f  2iw  2j(n- 

$  (aj  +  «„_))  =  -  s  MO  +  "2  cos  -*-  +  . . .  +  u.^_2  cos  - 


n         ) 


1    (  2?7T  4?7T 

-  n-z)  COS  -^  -  +  («g  +  ?<2n_4)  COS  -    - 


n  (  °  w  n 

I    (                JTT  3J7T  (2w-l)i7Tl 

I  (tt;  —  ttn_j)  =—•{«,  COS^ h  W3  COS  -+...  +  M2n_i  COS  — 

n  (  n  n  n         J 

1    f,                                    J7T  3J7T  ) 

=  -  S(MI  +  Wan-i)  COS1^ h  (U3  +  l^jn-s)  COS  —-  +  ...  X 

n  {  n  n  ) 

I  f       .    in-  .    3J7T  .    (2w-l)jV 

^  (o;  +  071-;)  =  -  -Ni  sml^ — I-  us  sm  —  -  +  ...  +  w^i  sm  - 

?i  (  n  n  n 


x    . 
-s)  sm  ^ 

/  t 


•  .  . 

sm  -  -  +  M4  sm  -  —  +  .  .  .  4-  w2»-2  sm 


- 

It     [  'ft/  it  if/ 

1  (,  2/7T  4iV         ) 

-  I  (Ms  -  ^2n-2)  Sin  -A-  +  ('M4  ~  U-zn-4)  BUI  -*j-  +  ... 


296-298]  Formulae  of  Numerical  Calculation 

(j  =  l,  2,  ...,  n  —  1);  and 


337 


n  =  -    "o 
71 


+  U3 


By  this  arrangement  an_/,  &„_/  are  calculated  together  with  o^,  6;  with  scarcely 
more  trouble  than  a/,  fy  alone.  As  a  practical  check  on  the  convergence  of 
the  series  these  higher  harmonics  should  be  found. 

298.     The  arrangement  can  be  greatly  simplified  in  special  cases.     For 
example,  in  the  case  s  =  12,  n  =  6,  let  the  data  be  arranged  thus  : 


Sums  :                 v0     v^      v2      v3     v4     v-a     v« 
Differences  :              wl     w2     w3     w4    w& 

V0       V-i        V2        V3 
V6       V5        V4 

Wl       W2      Ws 

Sums  :                p0     PI     p2     PS 
Differences  :       q0     ql     q2 

r-y»               (*• 
1          '2         '3 

a                ft 
Oi             Og 

The  equations  for  the  coefficients  are 

£  (a/  +  a6_j)  =  %(v0  +  v2  cos  \jir  +  vt  cos  |JTT  +  v6  cos  JTT) 
£  (a,—  a6_/)  =  ^  (v,  cos  |JTT  +  V3  cos  ^-JTT  +  vs  cos  f  JTT) 
^  (6/  +  66_j)  =  ^  (w!  sin  ^jir  +  w3  sin  ^;V  +  w6  sin  IJI'TT) 
^  (bj  —  b6-j)  =  ^  (w2  sin  ^jir  +  w4  sin  UTT). 

Hence  two  cases,  according  as  j  is  even  or  odd : 

j  even  j  odd 

:  (#0  +  <?a  cos  J  JTT) 

;  ^j  COS  IJTT 

:  (i*!  sin  ^JTT  +  rs  sin 
-  r2  sin  i  JTT 


,cos  *?7r) 

v     -v  A    "  j.     -•  O»/  / 

|  («j  —  a6_/)  =  £  (px  cos  iJ7r+ J»3  cos  \jtt 

=  i  *i  sin  i  jw 
^  (&>•  -  ^-j)  =  i  s2  sin  ^'TT 

and  these  forms  can  easily  be  made  more  general. 
p.  D.  A. 


22 


338  Formulae  of  Numerical  Calculation       [OH.  xxiv 

Then,  forj  =  2, 

I  (a.2  +  a4)  =  i  0>0  -  \p2\         £  (62  +  64)  =  i  *i  cos  30° 

£  (Oa  ~  <O  =  H  ipi  ~  PS),  i  (62  -  &4>  =  i  *a  COS  30° 

for  j  i  =  1  , 


(ax  +  a.)  = 

—  a5)  =  £  <?i  cos  30°,          i  (6j  —  68)  =  £  r2  cos  30° 
forj  =  3, 

«3  =  i  (?o  -  ^2),  &3  =  i  (n  -  r,) 

and  finally,  for  J  =  0, 


The  calculation  of  the  required  terms  is  therefore  extremely  simple.  The 
case  when  s=  24,  n—  12,  is  almost  equally  so,  but  would  require  more  space 
to  exhibit  in  detail. 

299.  The  mode  of  solution  for  the  harmonic  coefficients  can  be  con- 
sidered from  another  point  of  view.  Let  the  s  equidistant  values  u0,  ul,...,  u,^ 
be  given  as  before,  and  let  the  first  p  harmonics  —  including  ap,  bp  —  be 
required.  If  2p  =s  —  1,  the  number  of  unknowns  is  equal  to  the  number  of 
values  and  the  solution  is  unique.  If  1p  <  s  —  1,  the  number  of  equations  is 
in  excess  of  the  number  of  coefficients  to  be  determined.  The  latter  can 
then  be  found  by  the  rule  of  least  squares,  that  is,  so  as  to  make  the  sum  of 
the  squared  residuals  a  minimum.  The  equations  being  of  the  form 


P  (          Zirir      ,     .    2iW\ 

ur  =  a0  +  2,  [di  cos  —  —  +  6;  sin  -  ) 

i=i\  s  s    j 

the  quantity  which  is  to  be  made  a  minimum  is 
U  = 

r=o 

The  conditions  are 


(  £    /  2lW    ,     ,  2MTT\  |2 

\aQ  +  2,  [  a;cos  —  -  +  Offion—         -  ur\  . 

{          i=l\  S  S     J  } 


which,  being  2p  +  1  in  number,  determine  «0  and  the  2p  coefficients.     They 
give  in  fact 


V  (      ,    ^  f          2ir7r  ,  i.    • 
2,  4  a0  +  z  [cii  cos  ---  h  ft  Bin  —     -    —  ur  }•  =  0 

r=0  (  i-1  \  * 


• 


2  cos  -* —  \  a0  +  2  ( «i  cos h  b{  sin I 

r=o          *     (         i=i  \  s  s    / 

*~1    .      27>7T   f  ^    / 

2  sm-   -  ^a0+  2    a;: 

r=0  S       (  t-=1V 


;  cos  -       +    (-  sm 


298-soo]  Formulae  of  Numerical  Calculation  339 

But  since  2p<s  —  1,  0<j<p  +  l  and  0<i<p  +  I,  neither  i  nor  i+j  is 
a  multiple  of  s  (including  0).  Hence  the  only  terms  which  do  not  vanish  in 
the  sum  with  respect  to  r  arise  when  i—j=Q,  and  therefore  the  equations 
become  * 

s-l 

sa0  —  2  ur  =  0 

r=0 

•*^1  2?'r7r     _         ,      'z,1       .    2?r7r 

^Stty  —    2,   Mr  COS  —    -  =  0,       $SOj  —   2,   Ur  Sin  —     -  =  0 
r  =  0  S  r"=0  S 

(j  =  1,  . . . ,  p).  But  these  are  identical  with  the  earlier  equations  of  the  group 
(35)  when  the  distant  harmonics  are  omitted.  Hence  the  harmonics  to  any 
order  p  derived  by  the  general  rule  (36)  from  2n  equidistant  values  (p  <  n) 
are  the  same  as  would  result  from  a*  least-square  solution.  Thus  if  the 
function  is  represented  by  a  curve  and  the  coefficients  are  calculated  by  the 
rule,  «0  gives  the  best  horizontal  straight  line,  a0  +  aa  cos  9  +  6t  sin  6  the 
closest  simple  sine  curve,  and  so  on,  in  the  sense  denned.  This  important 
property  emphasises  the  independence  with  which  the  several  coefficients 
are  determined.  Each  apart  from  the  rest  is  found  with  the  greatest  possible 
accuracy  from  the  data  according  to  the  principle  of  least  squares. 

300.     The  method  can  be  extended  to  the  development  of  a  periodic 
function  in  two  variables, 

F=2aij  sin  (i0  +j0f  +  a). 
For  this  may  be  written  , 

F  =  a0  +  2  (ttj  cos  id  +  bi  sin  iO) 

i 

where  a0,  ai}  b{  are  each  of  the  same  form  as  F  with  &  in  the  place  of  6. 
With  any  particular  value  of  &  and  2n  equidistant  values  of  F  in  respect 
to  6,  a0,  (ii,  b{  can  be  determined  according  to  the  rule  expressed  by  (36).  Each 
of  these  is  a  function  of  the  chosen  value  of  #',  and  if  the  process  is  repeated 
with  2w  equidistant  values  of  6',  each  coefficient  can  be  expressed  in  the 
form 

dj  =  or0  +  2  (oii  cos  iff'  +  fa  sin  id') 

i 

by  the  same  rule.  When  these  expressions  are  inserted  in  the  second  form 
of  F,  the  first  form  is  readily  deduced.  This  method  was  employed  by 
Le  Verrier  in  his  theory  of  Saturn. 


(The  numbers  refer  to  pages.) 


Aberration,  91,  116,  117 
Absolute  perturbations,  180,  218 
Action,  136,  248 
Adams,  207,  258,  272 
Annual  equation,  282,  316 
Annual  precessions,  307 
Apollonius,  2 
Apparent  orbit,  81 
Appell,  165 
Apse,  Apsidal  angle,  6 
Argument  of  latitude,  65 
Arithmetic-geometric  mean,  161 
Astronomia  Nova,  1 
Astronomical  units,  19 
/3  Aurigae,  118 

Barker's  table,  26 

Bauschinger's  Tafeln,  26,  31,  32,  54,  58,  71, 

234 

Bernoulli,  D.,  48 
Bertrand,  5,  8 
Bessel,  37,  48,  327 

Bessel's  coefficients,  35,  36,  41,  42,  45—48 
Boys,  C.  V.,  10 
Braun,  K.,  10 
Brooks,  67 
Brown,  254,  279,  291 
Bruns,  15,  82,  215 
Burrau,  253 

Canonical  equations,  131,  152 

Cape  Observatory,  117 

Cassini's  laws,  312,  314,  315,  319,  320,  322 

Castor,  118 

Cauchy,  41,  159 

Cauchy's  numbers,  42,  43 

Cavendish,  10 

Cayley,  175 

Characteristic  exponents,  246,  271 

Characteristics,  order  of,  286 

Charlier,  76,  80,  81,  206 

Chrystal,  162 

Clairaut,  279 

Class  of  perturbation,  182,  191 

42  Comae  Berenices,  111 


Comet  a  1906,  67,  68 

Commensurability  of  mean  motions,  181,  191 

Conjugate  functions,  250,  258 

Contact  transformation,  132 

Continued  fraction,  162,  163 

Copernican  system,  1 

Cosmogony,  194 

Cowell,  173,  221 

Crommelin,  221 

Darboux,  6 

Darwin,  G.  H.,  238,  239,  264 

Degree  (of  perturbation),  182 

Delambre,  69,  100,  176 

Delaunay,  152,  153,  157,  175,  191,  254,  277, 

279, 285 
Descartes,  77 

Difference  table,  219,  324,  331 
Differential  corrections,  112,  126 
Disturbed  motion,  140,  243—245 
Disturbing  function,  19 
Diurnal  libration,  313 
Doppler,  115,  116 
Double  stars,  3,  19,  103 

Eccentric  anomaly,  3 

Eccentric  variables,  153 

Elements,  elliptic,  65 

of  double  stars,  104 

of  spectroscopic  binary,  121 

parabolic,  67 

Elimination  of  the  nodes,  186,  204 

Elliptic  functions,  159,  214,  253 

Eucke,  53,  64,  222 

Ephemeris,  75,  85,  323 

Equation  of  the  centre,  35,  40 

Eros,  206 

Euler,  48,  53,  96,  254,  260,  292,  313 

Eulerian  nutation,  295 

Evection,  279,  286 

Extended  point  transformation,  132,  266 

First  lunar  meridian,  315 

Fourier,  35,  40,  46,  121,  158,  261,  333,  334 

Franz,  316,  320 


342 


Gauss,  19,  31,  32,  69,  71,  85,  88,  89,  100,  162, 

207,  217,  326,  327 
Gaussian  constant,  20,  229 
Gegenschein,  242 
General  precession,  69,  302 
Geodetic  curvature,  82 
Gibbs,  62,  63,  91,  98 
Gravitation  constant,  10 
Green,  249 

Gudermannian  function,  27 
Gylden,  191,  242 

Halley's  comet,  221 

Halphen,  3,  6,  217 

Hamilton,  131,  134,  184 

Hamilton-Jacobi  equation,  133 — 135,  142,  146, 

154,  155,  188 

Hansen,  45,  167,  170,  191,  227,  254 
Hansen's  coefficients,  44,  46,  171,  174,  175 
Harmonic  analyser,  334 
Harmonic  analysis,  333 
Harmanicea  Mundi,  1 
Herschel,  J.,  107,  110,  125 
Herschel,  W.,  103 
Hessian,  202 
Hill,  G.  W.,  46,  217,  238,  245,  254,  258,  261, 

264—267,  269,  271,  272 
Hinks,  306 
Hodograph,  30 
Hypergeometric  series,  45,  159,  162,  165,  167, 

168,  215 

Inclination  of  orbit,  65 

Infinitesimal  contact  transformation,  139 

Integral  of  energy,  15,  16,  130,  131,  236,  260 

Integrals  of  area,  15,  185,  204 

Intermediate  orbit,  261 

Invariable  plane,  16,  17,  204 

Jacobi,  16,  164,  184,  186,  236 

Jupiter,  69,  164,  181,  191,  205,  224,  228,  234, 

235,  237,  243 
Jupiter  VIII  and  IX,  157,  222 

Kepler,  1,  2,  8—10,  111,  236,  315 
Kepler's  equation,  4,  24,  27,  29 — 31,  194 
Kinetic  focus,  136 
Klinkerfues,  82 
Kowalsky,  109 

Lagrange,  34,  46,  48,  74,  129,  130,  134,  200, 

244,  245,  328,  335 

Lagrange's  brackets,  136—138,  141,  144 
Lambert,  51,  55,  56,  81,  88 
Laplace,  17,  73,  190,  194,  203 
Laplace's  coefficients,  158—160,  169,  170,  174, 

.196—198 


Laurent,  40,  260,  261 
Least  action,  136 
Least  squares,  122,  338 
Legendre,  13,  165,  214,  215,  255 
Leonid  meteors,  207 
Le  Verrier,  164,  339 
Light  equation,  72,  91 
Limiting  curve,  79 
Locus  fictus,  71,  72 
Longitude  in  the  orbit,  65 
Longitude  of  perihelion,  65 
Long-period  inequalities,  181 
Lowell,  191 
Lunation,  284 
Luni-solar  precession,  300 

Major  planets,  164,  200,  218 
Mars,  1,  66,  205,  222 
Mass  of  Moon,  305,  306 
Mathieu's  equation,  246 
Mean  anomaly,  24 
Mean  longitude,  66,  153 
Mean  motion  of  node,  203 

of  perihelion,  201 

Mean  obliquity  of  ecliptic,  300,  302 
Mean  Sun,  308 
Mean  time,  308 

Mechanical  differentiation,  75,  329 
Mechanical  ellipticity  of  Earth,  305,  306 
Mehler-Dirichlet  integral,  214 
Mercury,  205 
Mimas,  191 
Minor  planets,   69,   102,   164,   191,   206,   228, 

243,  284,  306 
Motion  of  lunar  node,  285 

of  lunar  perigee,  279 
Moulton,  242 

Napier,  70 

Nautical  Almanac,  67,  68,  71,  72,  85,  228,  305, 

309 

Nebular  hypothesis,  194 
Neptune,  205,  235 

Newcomb,  20,  160—162,  164,  175,  307,  309 
Newcomb's  operators,  172,  173,  175 
Newton,  3,  9,  10,  25,  254,  325,  326 
Nodes,  65 

Nutational  ellipse,  303 
Nutation  constant,  304 

Oblique  variables,  153 
Olbers,  94 

Optical  libra tion,  313 
Order  of  perturbation,  182 
Osculating  orbit,  19,  178,  179 

Parallactic  inequality,  284 


Index 


343 


Parameter,  22 

Pascal,  106 

Periodic  orbits,  218,  238,  242,  243,  249,  261, 

264,  266 

Planetary  precession,  300 
Poincare,  H.,  15,  153,  159,  172,  182,  183,  191. 

246,  247,  261,  274 
Point  of  libration,  241 
Poisson,  140,  141,  190,  203,  322 
Poisson's   brackets,  134,   136—138,   140,  141, 

145,  146 
Polaris,  118 
Position  angle,  103 
Potential,  11 
Precession  constant,  304 
Principal  elliptic  term,  279,  316 
Principia,  3,  5,  7,  25 
Procyou,  114 

Protective  geometry,  104,  106 
Ptolemaic  system,  1 
Ptolemy,  279 
Puiseux,  322 

Quadrature,  218,  330 
Quaternions,  186 

Radial  velocity,  115 
Bank  of  perturbations,  182 
Relativity,  116 
Repulsive  forces,  27 
Resisting  medium,  177 
Retrograde  motion,  157,  194 

Satellite  motion,  157,  258 

Saturn,  181,  191,  205,  235,  339 

Schluter,  316 

Secular  acceleration  of  Moon,  291 

Secular  inequalities,  180 

Sidereal  time,  307 

Singular  curve,  80 


Sirius,  114 

Slipher,  191 

Special  perturbations,  218 

Spectroscopic  binaries,  115,  118 

Sphere  of  influence,  235 

Spiru-Haretu,  190 

Stability,  16,  180,  183,  190,  194,  199,  242,  243, 

246,  248,  271,  315,  319 
Steffensen,  267 
Stellar  kinematics,  117 
Stieltjes,  168 
Stirling,  326,  328 
Stockwell,  201,  205 
Stromgren,  253 

Taylor,  24,  171 
Theoria  Motns,  31,  32 
Thiele,  T.  N.,  107,  253 
Tisserand,  45,  168,  169,  237,  254 
Tropical  year,  310 
True  anomaly,  23 
Tycho  Brahe,  1,  2,  277 

Uranus,  205,  235 

Variable  proper  motion,  113 

Variation,  277 

Variational  curve,  261,  266,  267 

Variation  of  constants,  134 

Variation  of  latitude,  295 

Velocity  curve,  118 

Venus,  205,  222 

Weierstrass,  159,  200,  214 
Whittaker,  46,  48,  214,  215,  248,  269 
Whittaker    and  Watson,   46,   214,    215,    247, 
269 

Zeipel,  H.  von,  158,  164,  207 
Zwiers,  107 


CAMBRIDGE  :   PRINTED  BY  j.  B.  PEACE,  M.A.,  AT  THE  UNIVERSITY  PKESS. 


FOURTEEN  DAY  USE 

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t-JBp^TlH- 

JUN29  1961 

Rwcd  UC6  A/M/S 

AI  i  r*   f\        «j-\.  J-L  j 

AUG  3     1981 

CD37StE7clfl 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

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